C$^*$-simple groups: \\ Amalgamated free products, \\ HNN extensions, \\ and fundamental groups of $3$-manifolds
Pierre de la Harpe and Jean-Philippe Pr\'eaux
We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups.~The second class is that of fundamental groups of compact $3$-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompo\-sitions. \par Much of our analysis deals with conditions on an action of a group $\Gamma$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree $T$ and on its boundary $\partial T$, and faithfulness in a strong sense. For this, we define in particular the notion of a slender automorphism of $T$, namely of an automorphism such that its set of fixed points on $\partial T$ is nowhere dense with respect to the shadow topology.
0909.3528.tex

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\newtheorem{Exam}[Def]{Example}
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\title[C$^*$-simple groups]
{C$^*$-simple groups: \\
Amalgamated free products, \\
HNN extensions, \\
and fundamental groups of $3$-manifolds
}



\author{Pierre de la Harpe
and
Jean-Philippe Pr\'eaux
}

\thanks{The authors acknowledge support 
from the Swiss national science foundation}

\address{Pierre de la Harpe~:
Section de math\'ematiques, 
Universit\'e de Gen\`eve, 
C.P.~64, 
CH--1211 Gen\`eve 4. 
}
\email{Pierre.delaHarpe@unige.ch}


\address{
Jean-Philippe Pr\'eaux~:
Centre de recherche de l'arm\'ee de l'air,
Ecole de l'air/BA701,
F-13661 Salon-air~; 
\newline
and:
Laboratoire d'analyse topologie et probabilit\'es,
Universit\'e de Provence,
39 rue F. Joliot-Curie,
F-13453 Marseille cedex 13.
}

\email{preaux@cmi.univ-mrs.fr}

\keywords{Groups acting on trees, slender automorphism, 
amalgamated free products, HNN extensions, 
Baumslag-Solitar groups, graphs of groups, 
fundamental groups of $3$-manifolds,
C$^*$-simplicity.}
\subjclass[2000]{22D25, 20E08, 20F65, 57N10}
% 22D25 = C*-algebras and W*-algebras in relation to group representations
% 20E08 = group acting on trees
% 20F65 = geometric group theory
% 57N10 = topology of general 3-manifolds

\date{September 18, 2009}

\begin{document}

\begin{abstract} 
We establish sufficient conditions for the C$^*$-simplicity
of two classes of groups.
The first class is that of groups acting on trees, 
such as amalgamated free products, HNN-extensions, and their normal subgroups;
for example normal subgroups of Baumslag-Solitar groups.~The 
second class is that of fundamental groups
of compact $3$-manifolds, 
related to the first class by their 
Kneser-Milnor and JSJ-decompo\-sitions.
\par
Much of our analysis deals with conditions 
on an action of a group $\Gamma$ on a tree $T$
which imply  the following three properties:
abundance of hyperbolic elements, 
better called strong hyperbolicity,
minimality, 
both on the tree $T$ and on its boundary $\partial T$,
and faithfulness in a strong sense.
For this, we define in particular the notion 
of a \emph{slender} automorphism of  $T$,
namely of an automorphism such that 
its set of fixed points on $\partial T$
is nowhere dense with respect to the shadow topology.
\end{abstract}

\maketitle

\section{Introduction
}
\label{section1}

In the first part of this paper, we analyse actions of groups on trees,
and we  establish that the reduced C$^*$-algebras 
of some of these groups are simple.
In the second part, we apply this to 
fundamental groups of compact $3$-manifolds
and their subnormal subgroups.

Given a group $\Gamma$, recall that its \emph{reduced C$\,{}^*$-algebra}
$C^*_r(\Gamma)$ is the closure for the operator norm
of the group algebra $\mathbf C [\Gamma]$ 
acting by the left-regular representation 
on the Hilbert space $\ell^2(\Gamma)$.
For an introduction to group C$^*$-algebras,
see for example Chapter VII of \cite{Davi--96}.
A group is \emph{C$\,{}^*$-simple} if 
it is infinite and if
its reduced C$^*$-algebra 
has no non-trivial two-sided ideals. 
\par

Non-abelian free groups are C$^*$-simple.
The first proof, due to Powers \cite{Powe--75}, 
relies on a combinatorial property of free groups
shared by many other groups,
called for this reason \emph{Powers groups};
the definition is recalled in Section \ref{section7} below.
For now, we emphasize that
\begin{center}
\emph{Powers groups are C$\,{}^*$-simple}
\end{center}
(the converse does \emph{not} hold).
Whenever a group is known to be a Powers group,
the proof involves  Proposition \ref{Powersgen} below,
or a variant thereof.
Let us agree that a 
\begin{itemize}
\item[$\bullet$]
\emph{strongly Powers group} is a group $\Gamma$ 
such that any subnormal subgroup $N \ne \{1\}$
of $\Gamma$ is a Powers group.
\end{itemize}
(The definition of ``subnormal'' is recalled in the proof
of Proposition \ref{Powerspart}.)  % 18
More on these groups in Section \ref{section7} and in \cite{Harp--07}.

\bigskip

Part I, on groups acting on trees, 
begins with Section \ref{section2}, 
where we collect essentially known facts
which are useful for the proof of Proposition \ref{Powerstree}. % 1
We need first some terminology.
A \emph{pending ray} in a tree $T$ is a ray with vertex set
$\left(x_n\right)_{n \ge 1}$ such that  $x_n$ has degree $2$ in $T$
for all $n$ large enough.
Cofinal classes of rays are the elements 
of the \emph{boundary} $\partial T$,
which is a  space with a natural \emph{shadow topology}
(more on this in Section \ref{section2}).
An action of a group $\Gamma$ on  a tree $T$ is 
\begin{itemize}
\item[$\bullet$]
\emph{minimal}
if there does not exist any proper $\Gamma$-invariant subtree in $T$.
\end{itemize}
If $\Gamma$ acts by automorphisms on $T$, 
it acts also by homeomorphisms on $\partial T$.
It is important not to confuse 
the minimality of the action on the tree
and the minimality of the action on the boundary
(which means that the boundary does not have
any non-trivial $\Gamma$-invariant closed subspace).
An automorphism $\gamma$ of $T$, with $\partial T \ne \emptyset$, is 
\begin{itemize}
\item[$\bullet$]
\emph{slender}
if its fixed point set in $\partial T$ has empty interior,
\end{itemize}
and the action of $\Gamma$ on $T$ is \emph{slender}
if any $\gamma \in \Gamma$, $\gamma \ne 1$, is slender.
Observe that a slender action is faithful.
As our trees are assumed to be non-empty,
observe also that a tree on which a group acts 
without fixed vertex and without fixed geometric edge
has infinite diameter and non-empty boundary.

\begin{Prop}
% 1
\label{Powerstree}
Let~$T$ be a tree without vertices of degree $\le 1$ 
and without pending rays.
Let $\Gamma$ be a countable group which acts on $T$.
We assume that the action is 
minimal (in particular without fixed vertex),
without fixed boundary point, and slender.
\par

Then $\Gamma$ is a strongly Powers group.
\end{Prop}

Section \ref{section3} is a reminder 
on  the fundamental group $\Gamma$
of a graph of groups $\mathbf G  = (G,Y)$,
which acts on its Bass-Serre tree $T$.
We recall known criteria for $T$ to be ``small''
(a vertex, or a linear tree),
for the action to be faithful, 
and for the action to be minimal.
Most of this can be found in papers by H.~Bass.
In Section \ref{section4}, we analyse further
the two standard examples
and obtain the following results;
they rely on Proposition \ref{Powerspart}, % 18
which is a reformulation of Proposition \ref{Powerstree}. % 1
\par

Let $A,B,C$ be three groups,
let $\iota_A, \iota_B$ be injections of $C$ in $A, B$ respectively,
and let
\begin{equation*}
\Gamma \, = \, A \ast_C B \, = \, 
\langle A,B \hskip.1cm \vert \hskip.1cm \iota_A(c) = \iota_B(c)
\hskip.2cm \forall c \in C \rangle
\end{equation*}
be the corresponding amalgam.
Below, we identify $C$ to a subgroup of $A$ and of $B$.
We define inductively a decreasing sequence 
$C_0 \supset C_1 \supset \cdots$ of subgroups of $C$
by $C_0 = C$ and 
\begin{equation*}
C_{k} \, = \, 
\big( \bigcap_{a \in A} a^{-1}C_{k-1} a \big)  
\cap
\big( \bigcap_{b \in B} b^{-1}C_{k-1} b \big)
\hskip.5cm \text{for} \hskip.2cm k \ge 1 .
\end{equation*}


\begin{Prop}
% 2
\label{C*simpleamalg}
Consider as above a countable amalgam
$\Gamma \, = \, A \ast_C B$,
with moreover $[A : C] \ge 3$ and $[B : C] \ge 2$.
Assume that
$C_k \, = \, \{1\}$ for some $k \ge 0$.
\par
Then $\Gamma$ is a strongly Powers group.
\end{Prop}

The condition $C_k = \{1\}$ has the following geometrical interpretation:
for $\gamma \in \Gamma$, 
if there exists $e \in E(T)$ such that
the $k$-neighbourhood
\begin{equation*}
\mathcal V_k(e) \, = \,
\{ x \in V(T) \hskip.1cm \vert
\min \{ d(x, s(e)), \hskip.1cm d(x,t(e)) \} \le k \}
\end{equation*}
is pointwise fixed by $\gamma$, then $\gamma = 1$.
In particular, if $C_k = \{1\}$ for some $k \ge 0$, 
then the action of $\Gamma$ on $T$ is faithful.
\par

Proposition \ref{C*simpleamalg} % 2
is an improvement on previous results in two ways.
(i)~It establishes a property of any subnormal subgroup $N \ne \{1\}$, 
and not only of $N = \Gamma$.
(ii)~Its hypothesis are stated in terms of $A$, $B$, and $C$ only;
on the contrary, 
with the same conclusion ``$\Gamma$ is a Powers group'',
Proposition 10 in \cite{Harp--85} and Corollary 4.6 in \cite{Ivan}
have hypothesis stated in terms
of the action of $\Gamma$ on the edge set of its Bass-Serre tree
(more precisely, 
this action should be strongly faithful, as defined in Section~\ref{section2}).
Our conditions in Proposition \ref{C*simpleamalg} % 2
are also weaker than those of \cite{Bedo--84}.
\par
A similar remark holds for Proposition \ref{C*simpleHNN} below and
Proposition 11 in \cite{Harp--85}.
\par
% 
A particular case of Proposition \ref{C*simpleamalg} is well-known:
the  case of free products  \cite{PaSa--79}.
A free product $A \ast B$ is \emph{non-trivial}
if neither $A$ nor $B$ is the group with one element.
Recall that the \emph{infinite dihedral group}
is the free product of two groups of order two,
and that it is the only non-trivial amenable free product.


\begin{Cor}
% 3
\label{Freeproducts}
A non-trivial free product $A \ast B$
is C$\, ^*$-simple if and only if it is not isomorphic
to the infinite dihedral group.
\end{Cor}

At first sight, Corollary \ref{Freeproducts} % 3
follows from Proposition \ref{C*simpleamalg} % 2
for countable groups only.
But a group $\Gamma$ is C$^*$-simple 
[respectively Powers, strongly Powers] as soon as,
for any countable subgroup $\Gamma_0$ of $\Gamma$,
there exists a countable C$^*$-simple 
[respectively Powers, strongly Powers] subgroup
of $\Gamma$ containing $\Gamma_0$;
see Proposition 10 in \cite{BeHa--00} and Lemma 3.1 in \cite{BoNi--88}.
It follows that Corollary~\ref{Freeproducts}
holds as stated above
[and also with ``Powers group'' or ``strongly Powers group''
instead of  ``C$^*$-simple''].
\par



Let $G$ be a group, let $\theta$ be an isomorphism 
from a subgroup $H$ of $G$ to some subgroup of $G$,
and let
\begin{equation*}
\Gamma \, = \,  HNN(G, H, \theta)
\, = \,
\langle G, \tau \hskip.1cm \vert \hskip.1cm
\tau^{-1}h\tau = \theta(h) \hskip.2cm
\forall h \in H \rangle 
\end{equation*}
be the corresponding HNN-extension.
We define inductively a decreasing sequence
$H_0 \supset H_1 \supset \cdots$ of subgroups of $H$
by $H_0 = H$ and
\begin{equation*}
\aligned
H'_k \, &= \, H_{k-1} \cap \tau^{-1} H_{k-1} \tau \, = \, 
H_{k-1} \cap \theta( H_{k-1} )
\\
H_k \, &= \, 
\Big( 
    \bigcap_{g \in G} g  H'_{k-1}  g^{-1} 
\Big)  
\cap
\tau \Big( 
    \bigcap_{g \in G} g  H'_{k-1}  g^{-1} 
\Big)   \tau^{-1}
\endaligned
\end{equation*}
for $k \ge 1$.


\begin{Prop}
% 4
\label{C*simpleHNN}
Consider as above a countable HNN-extension 
$\Gamma =  HNN(G, H, \theta)$,
with moreover $H \subsetneqq G$ and $\theta(H) \subsetneqq G$.
Assume that $H_k = \{1\}$ for some $k \ge 0$.
\par

Then $\Gamma$ is a strongly Powers group.
\end{Prop}

As for Proposition \ref{C*simpleamalg}, % 2
the condition $H_k = \{1\}$ has the following geometrical interpretation:
for $\gamma \in \Gamma$, 
if there exists $e \in E(T)$ such that
the $k$-neighbourhood $\mathcal V_k(e)$
is pointwise fixed by $\gamma$, then $\gamma = 1$.
\par

Though it does not quite follow from Proposition \ref{C*simpleHNN},
similar arguments imply the next proposition.
Its first part is due to Nikolay Ivanov, 
with a different proof (Theorem 4.9 in  \cite{Ivan}).
In the second part,  for $p \ge 0$,
we denote abusively by $\theta^p(H)$ the image
of the restriction of $\theta^p$ to
$\{h \in H \hskip.1cm \vert \hskip.1cm \theta^j(h) \in H
\hskip.2cm \text{for} \hskip.2cm j = 1, \hdots, p-1 \}$,
and similarly for $\theta^{-p}(H)$.

\begin{Prop}
% 5
\label{BS}
The Baumslag-Solitar group
\begin{equation*}
BS(m,n) \, = \, 
\langle \tau,b \hskip.1cm \vert \hskip.1cm \tau^{-1}b^m\tau = b^n \rangle
\, = \,
HNN(b^{\mathbf Z}, b^{m\mathbf Z}, \hskip.1cm
b^{mk} \longmapsto b^{nk})
\end{equation*}
is a strongly Powers group if and only if  it is C$\,^*$-simple,
 if and only if \hfill \par\noindent
$\min\{\vert m\vert , \vert n \vert\} \ge 2$
and $\vert m \vert \ne \vert n \vert$.
\par

More generally, 
let $\Gamma$ be a countable group
which is an HNN-extension $HNN(G,H,\theta)$ such that
$H \subsetneqq G$,  $\theta(H) \subsetneqq G$,
and at least one of
\begin{equation*}
K_+ \, \Doteq \,  \bigcap_{p \ge 0} \theta^p (H) \, = \, \{1\} ,
\hskip.5cm  
K_+ \, \Doteq \, \bigcap_{p \ge 0} \theta^{-p}(H) \, = \, \{1\}  
\end{equation*}
holds.
Then $\Gamma$ is a strongly Powers group.
\end{Prop}

Consider for example 
the homomorphism $\sigma : BS(m,n) \longrightarrow \mathbf Z$
defined by $\sigma(b) = 0$ and $\sigma(\tau) = 1$,
and the \emph{special Baumslag-Solitar group}
\begin{equation*}
SBS(m,n) \, = \, \ker (\sigma) .
\end{equation*}
We have:


\vskip.2cm

\begin{Cor}
% 6
\label{SBS}
If $\min\{\vert m \vert, \vert n \vert\} \ge 2$
and $\vert m \vert \ne \vert n \vert$,
then $SBS(m,n)$ is a strongly Powers group.
\end{Cor}


The subject of Part II is the fundamental group $\Gamma$
of a  connected compact $3$-manifold $M$
(with or without boundary).
Let $\widehat M$ be the manifold
obtained from $M$ by filling all $2$-spheres in $\partial M$ 
with $3$-balls
(thus $M$ is the connected sum of $\widehat M$
and as many $3$-balls as the number of 
components of $\partial M$ which are $2$-spheres).
Since $\pi_1(\widehat M) = \pi_1(M)$,
it is often convenient to replace $M$ by $\widehat M$.
For background on $3$-manifolds, 
we refer to  \cite{Hemp--76}, \cite{JaSh--79}, 
\cite{Scot--83},  \cite{Thur--97}, and \cite{Bona--02};
see also Section \ref{section5}.
\par

Let us first describe four classes of examples
for which it is easy to see wether $\Gamma$ is C$^*$-simple or not.
\par

(i)
A \emph{Seifert manifold} is a connected $3$-manifold 
which can be foliated by circles, usually called \emph{fibers},
such that each fiber has a neighbourhood, 
either a solid torus or a solid Klein bottle,
which is  a union of fibers in some standard way
(``such that ..'' is automatic for a compact manifold,
by a theorem of David Epstein).     
Let $M$ be a  Seifert manifold 
with infinite fundamental group $\Gamma$
(equivalently: which is not covered by a $3$-sphere,
see Lemma 3.2 in \cite{Scot--83});
if $M$ is given a Seifert structure,
a generic fiber generates a normal subgroup of $\Gamma$ 
which is infinite cyclic.
Hence $\Gamma$ is not C$^*$-simple,
because a C$^*$-simple group
cannot have non-trivial amenable normal subgroups
(Proposition 3 of \cite{Harp--07}).
\par

(ii)
A \emph{Sol-manifold} is a connected $3$-manifold $M$
of which the interior $\mathring{M}$ can be given a 
Riemannian structure
such that the universal covering 
$\pi : Sol \longrightarrow \mathring{M}$ is a local isometry.
Here, $Sol$ denotes the $3$-dimensional Lie group
with underlying space $\mathbf R^3$,
with product
\begin{equation*}
(x,y,z)(x',y',z') \, = \, (x + e^{-z}x' , y + e^zy' , z + z') ,
\end{equation*}
and with Riemannian structure
\begin{equation*}
ds^2 \, = \,  e^{2z}dx^2 + e^{-2z}dy^2 + dz^2 .
\end{equation*}
The fundamental group of a Sol-manifold
contains a subgroup of finite index $\Gamma_0$
which fits in a short exact sequence
\begin{equation*}
\{1\} \, \longrightarrow \, 
K  \, \longrightarrow \, 
\Gamma_0  \, \longrightarrow \, 
Q  \, \longrightarrow \,  \{1\} 
\end{equation*}
where $K$ [respectively $Q$]
is a discrete subgroup of the group of isometries
of a Euclidean plane [respectively of  a line];
see Proposition 4.7.7 in \cite{Thur--97}.
In particular, $\Gamma$ has a solvable subgroup of finite index, 
hence $\Gamma$ is amenable, 
and therefore $\Gamma$ is not C$^*$-simple.
\par


(iii)
Let $M$ be a connected $3$-manifold which is
hyperbolic and not elementary
(for example of finite volume).
If $M$ is orientable,
then $\Gamma$ is isomorphic to 
a subgroup of $PSL_2(\mathbf C)$ which is not almost solvable,
and these are  strongly Powers groups.
(See Section \ref{section7}; see also
Proposition 6 in \cite{Harp--85},
which could be added to the list
of Corollary 12 in \cite{Harp--07}.)
If $M$ is non-orientable,
the orientation subgroup $\Gamma'$ of $\Gamma$,
which is of index two, is C$^*$-simple, by the previous argument;
since $\Gamma$ is infinite 
and has infinite conjugacy classes (Lemma 9.1 in \cite{HaPr--07}),
$\Gamma$ itself is C$^*$-simple by \cite{BeHa--00}.
\par


(iv)
If the compact $3$-manifold $\widehat M$ is not prime
(= is a connected sum in a non-trivial way), 
and if we assume the Poincar\'e Conjecture,
then $\Gamma$ is a non-trivial free product.
Suppose that, moreover, $M$ is orientable.
If $\Gamma$ is an infinite dihedral group, then $\widehat M$ is Seifert;
in all other cases, $\Gamma$ is a strongly Powers group,
by Proposition \ref{C*simpleamalg}. % 2
(More on this in the discussion of Section~\ref{section5}.)
\par

We sum up the previous discussion as follows:

\begin{Exam}
% 7
\label{Ex1a4}
Let $M$ be a connected  compact $3$-manifold !
and let $\Gamma$ denote its fundamental group.
\begin{itemize}
\item[(i)] 
If $\widehat M$ is a Seifert manifold, $\Gamma$ is not C$\, ^*$-simple.
\item[(ii)] 
If $\widehat M$ is a Sol-manifold, $\Gamma$ is not C$\, ^*$-simple.
\item[(iii)] 
If $\widehat M$ is hyperbolic and non-elementary, $\Gamma$ is  C$\, ^*$-simple.
\end{itemize}
Suppose moreover that $M$ is orientable, 
and that the Poincar\'e Conjecture is true.
\begin{itemize}
\item[(iv)] 
If $\widehat M$ is not prime and not Seifert, $\Gamma$ is 
a strongly Powers group.
\end{itemize}
\end{Exam}



In a sense,  the existence of a Seifert structure
and that of a Sol-structure on $\widehat M$
are the only obstructions to the C$^*$-simplicity of $\Gamma$.
More precisely, 
as a consequence of Propositions \ref{C*simpleamalg} % 2 
and \ref{C*simpleHNN}, % 4
and as we show in Section \ref{section6}:

\begin{Prop}
% 8
\label{orientable} 
Let $M$ be an orientable connected compact $3$-manifold
and let $\widehat M$ be as above.
We assume that Thurston's Geometrisation Conjecture holds.
\par
If $\widehat M$ is not a Seifert manifold and is not a Sol manifold,
then the fundamental group $\Gamma = \pi_1(M)$ is 
a strongly Powers group.
\end{Prop}


\begin{Cor}
% 9
\label{knot}
Let $K$ be a knot in $\mathbf S^3$ 
and let $\Gamma$ denote the fundamental group of its complement.
\par
Then $\Gamma$ is a strongly Powers group if and only if $K$
is not a torus knot.
\end{Cor}

For non-orientable manifolds, we will restrict ourselves
to the following simple statement:

\begin{Prop}
% 10
\label{nonorientable} 
Let $M$ be a non-orientable connected compact  $3$-manifold, 
$\Gamma$  its fundamental group,
and $\Gamma'$  the fundamental group 
of the total space of the orientation cover of $M$;
in particular,  $\Gamma'$ is a subgroup of index $2$ in $\Gamma$.
\par

Then $\Gamma$ is C$\,{}^*$-simple if and only if 
$\Gamma'$ is C$\,{}^*$-simple.
\end{Prop}


\emph{Remark on groups of  surfaces and groups of $3$-manifolds.} 
Let $S$ be a connected compact surface.
It follows from the classification of surfaces and from elementary arguments
(compare with \cite{HaPr--07}) that the following three properties are equivalent:
\begin{itemize}
\item[(i)]
$S$ is not homeomorphic to a disc, a sphere, a projective plane, 
an annulus, a M\"obius band, a $2$-torus, or a Klein bottle; 
\item[(ii)]
$\Gamma$ is icc (namely all its conjugacy classes
other than $\{1\}$ are infinite);
or equivalently the von Neumann algebra of $\Gamma$
is a factor of type $II_1$;
\item[(iii)]
$\Gamma$ is C$^*$-simple.
\end{itemize}
If $\Gamma$ is now the fundamental group 
of a connected compact $3$-manifold,
the equivalence of (ii) and (iii) does not carry over.
Indeed, the fundamental group of a Sol manifold can be icc,
and is never C$^*$-simple.
In some sense, Propositions \ref{orientable} % 8
and \ref{nonorientable} % 10
provide the $3$-dimensional analogue of the equivalence
$(i) \Longleftrightarrow (iii)$.


\vskip.2cm

We are grateful to Luc Guyot for suggesting 
that most of our propositions establish not only that 
some $\Gamma$ is a Powers group,
but also that any subnormal subgroup $N \ne \{1\}$ 
is a Powers group,
and for many other comments
on  preliminary versions of this paper.
We are also grateful to
Laurent Bartholdi, Bachir Bekka, Ken Dykema,  
Yves Stalder, and Nicolas Monod 
for  helpful remarks.

\newpage

\begin{center}
\textbf{I. Fundamental groups of graphs of groups}
\end{center}


\section{On groups acting on trees
}
\label{section2}

Let $X$ be a \emph{graph}, with vertex set $V(X)$ and edge set $E(X)$.
We follow Bass \cite{Bass--93} and Serre \cite{Serr--77};
in particular, each geometric edge of $X$
corresponds to a pair $\{e,\overline{e}\} \in E(X)$,
with $\overline{e} \ne e$ and $\overline{\overline{e}} = e$.
Each edge $e \in E(X)$ has a \emph{source} $s(e) \in V(X)$ and a 
\emph{terminus}, $t(e) \in V(X)$.
We denote by $d(x,y)$ the \emph{combinatorial distance}
between two vertices $x,y \in V(X)$.
In case of a \emph{tree}, we write $T$ rather than $X$,
and we agree that $V(T) \ne \emptyset$.
\par

We denote by $\partial T$ the  \emph{boundary} of a tree $T$.
Recall that a \emph{ray} in $T$ is a subtree 
with vertex set $(x_n)_{n \in \mathbf N}$, such that
\begin{equation*}
d(x_m,x_n) \, = \,  \vert m-n \vert \hskip.5cm
\text{for all} \hskip.2cm m,n \in \mathbf N.
\end{equation*}
The enumeration of the vertices of a ray will \emph{always}
be such that the above condition holds.
Two rays with vertex sets 
$(x_n)_{n \in \mathbf N}$ and $(y_n)_{n \in \mathbf N}$
are \emph{cofinal} if there exists some $k \in \mathbf Z$
such that $y_n = x_{n+k}$ for all $n$ large enough,
and  $\partial T$ is the set of cofinal classes of rays in $T$.

The set $\partial T$ has a natural topology defined as follows.
For any edge $e \in E(T)$, 
the \emph{shadow} $(\partial T)_e$ of $e$ in $\partial T$
is the subset of $\partial T$
represented by rays $(x_n)_{n \in \mathbf N}$
such that $d(t(e),x_n) < d(s(e),x_n)$ for all $n$ large enough.
The family of shadows $\left\{ (\partial T)_e \right\}_{e \in E(T)}$
generates a topology on $\partial T$
which is Hausdorff and totally disconnected;
if $T$ is countable, this topology is moreover metrisable.
Given any  $x_0 \in V(T)$,
this topology coincides\footnote{
%
Similarly, one can define two topologies on the disjoint union
$\overline{T} = V(T) \sqcup \partial T$,
one using appropriate shadows and 
one by identifying $\overline{T}$ 
to the inverse limit of the balls
$\{x \in V(T) \hskip.1cm \vert \hskip.1cm d(x_0,x) \le n\}$.
These two topologies coincide if and only if
the tree $T$ is locally finite,
but their restrictions on $\partial T$ coincide in all cases.
The shadow topology makes $\overline{T}$ a compact space,
but $\partial T$ need not be closed,
and therefore need not be compact
(unless $T$ is locally finite). More on this in \cite{MoSh--04}.
}
%
with the inverse limit topology
on the set $\partial T$,  
identified with the inverse limit of the discrete spaces 
$S(x_0,n) = \{x \in V(T) \hskip.1cm \vert \hskip.1cm d(x_0,x) = n\}$, 
namely of the \emph{spheres} in $T$ around $x_0$.
It is known that $\partial T$ is compact if and only if $T$ is locally finite
(and we do consider below trees \emph{which are not} locally finite).
See the last exercise in Section I.2.2 of \cite{Serr--77},
Section I.8.27 in \cite{BrHa--99},
and Section 4 in \cite{MoSh--04}.


Observe that, for $\partial T$ to be non-empty,
it suffices that $T$ has more than one vertex
and does not have any  vertex of degree $1$.
\par


A ray cofinal with a pending ray is also pending.
The classes of the pending rays are precisely 
the isolated points in $\partial T$;
in particular, the topological space $\partial T$
is \emph{perfect} if the tree $T$ does not have any pending ray.
(Pending rays are defined before Proposition \ref{Powerstree}.) % 1


\begin{Prop}
% 11
\label{Baire}
For any countable tree $T$, the boundary $\partial T$ is a Baire space.
\end{Prop}

\emph{Proof.}
Choose a vertex $x_0 \in V(T)$.
Since $T$ is countable, 
each of the spheres $S(x_0,n)$ defined above is countable,
and therefore is a polish space for the discrete topology.
Since the inverse limit of a countable inverse system of polish spaces 
is polish (\cite{Bou-TG9}, $\S$ 6),
$\partial T$ is polish
(``countable'' is important, because a product 
of uncountably many polish spaces is not polish in general).
Since polish spaces are  Baire spaces 
(\cite{Bou-TG9}, $\S$~5),
this ends the proof.
\hfill $\square$

\vskip.2cm

Recall that an action of a group $\Gamma$ on a set $\Omega$ is
\begin{itemize}
\item[$\bullet$]
\emph{faithful} if, 
for any $\gamma \in \Gamma$, $\gamma \ne 1$, 
there exists $\omega \in \Omega$ such that
$\gamma(\omega) \ne \omega$, and
\item[$\bullet$]
\emph{strongly faithful} if, 
for any finite subset $F$ of $\Gamma$ not containing $1$,
there exists $\omega \in \Omega$ such that 
$\gamma (\omega) \ne \omega$ for all $\gamma \in F$.
\end{itemize}
An action of a group on a tree $T$ is faithful, or strongly faithful,
if it is the case for the action on the vertex set $V(T)$.
% 
Let  $\Omega$ be a Hausdorff space 
and let the action of $\Gamma$ be by homeomorphisms;
if the action is strongly faithful 
and if $F$, $\omega$ are as above,
observe that there exists 
a neighbourhood $V$ of $\omega$ in $\Omega$
such that $\gamma (V) \cap V = \emptyset$ 
for all $\gamma \in F$.
Recall also from the introduction that an action of a group $\Gamma$
on a tree $T$ is
\begin{itemize}
\item[$\bullet$]
\emph{slender} if any $\gamma \in \Gamma$, $\gamma \ne 1$,
is slender, namely is such that
the fixed point set $(\partial T)^{\gamma}$ has empty interior.
\end{itemize}

\begin{Cor}
% 12
\label{slender}
Let $\Gamma$ be a countable group which acts by automorphisms 
on a tree $T$.
\par
If the action on $T$ is slender, 
then the action on $\partial T$ is strongly faithful.
\end{Cor}

\noindent 
\emph{Proof.}~
Let $F$ be a finite subset of $\Gamma \smallsetminus \{1\}$,
as in the definition of ``strongly faithful''.
Since the action is slender,
the fixed point set $(\partial T)^\gamma$ has no interior 
for all $\gamma \in F$, 
so that there exists by Proposition \ref{Baire}
(Baire's property of $\partial T$)
a point $\xi \in \partial T$
such that $\gamma(\xi) \ne \xi$ for all $\gamma \in F$.

[Observe that
$\partial T \smallsetminus 
\bigcup_{\gamma \in \Gamma \smallsetminus \{1\}} (\partial T)^{\gamma}$
is dense, and therefore non-empty, 
so that there exists $\xi \in \partial T$
such that $\gamma(\xi) \ne \xi$ 
for all $\gamma \in \Gamma \smallsetminus \{1\}$.
We shall not use this below.]
\hfill $\square$

\vskip.2cm

Since \cite{Tits--70}, we know that
an automorphism $\gamma$ of a tree $T$
can be of three different kinds:
\begin{itemize}
\item[$\bullet$]
It is an \emph{inversion} if there exists $e \in E(T)$
such that $\gamma(e) = \overline{e}$.
\item[$\bullet$]
It is \emph{elliptic} if it has at least one fixed point in $V(T)$.
\item[$\bullet$]
It is \emph{hyperbolic} if it is not an inversion and if 
\begin{equation*}
d(\gamma) \Doteq \min\{d(y,\gamma(y)) 
\hskip.1cm \vert \hskip.1cm y \in V(T) \} \ge 1.
\end{equation*}
If $\gamma$ is hyperbolic, it has an \emph{axis},
which is a linear subtree $T_{\gamma}$, such that  
$x \in V(T_{\gamma})$ if and only if
$d(x,\gamma(x)) = d(\gamma)$,
and $d(\gamma)$ is called the \emph{translation length} of $\gamma$.
\end{itemize}
We translate here by 
\begin{itemize}
\item[$\bullet$]
\emph{linear subtree} 
\end{itemize}
what is a ``cha\^{\i}ne'' in \cite{Tits--70}
and a ``droit chemin'' in \cite{Serr--77},
namely a subtree of which the vertex set
is of the form $(x_n)_{n \in \mathbf Z}$,
with $d(x_m,x_n) = \vert m - n \vert$ for all $m,n \in \mathbf Z$.
Linear trees are defined accordingly.

\begin{Rem}
% 13
\label{Remarks}
(i)
An automorphism $\gamma$ of $T$ which is hyperbolic
has exactly two fixed points on $\partial T$
which are its \emph{source} $\alpha(\gamma)$
and its \emph{sink} $\omega(\gamma)$,
and the infinite cyclic group $\gamma^{\mathbf{Z}}$
acts freely on 
$\partial T \smallsetminus \{\alpha(\gamma), \omega(\gamma) \}$.
If $\gamma$ is an hyperbolic automorphism of a tree
of which the boundary consists of more than two points
(and therefore of which the boundary is infinite), 
then $\gamma$ is always slender, 
since its fixed point set in $\partial T$ 
consists of two non-isolated points.
\par
(ii)
If $T$ is a tree with non-empty boundary $\partial T$
and if $\gamma$ is an elliptic automorphism of $T$
with fixed point set $V(T)^{\gamma}$ of finite diameter,
then $(\partial T)^{\gamma} = \emptyset$,
and in particular $\gamma$ is slender.
\par
(iii)
If $T$ is the Bass-Serre tree corresponding 
to the Baumslag-Solitar group $\Gamma = BS(m,n)$ 
for some $m,n \in \mathbf Z$
(as in Subsection \ref{ssHNN} below), 
there exist  elliptic automorphisms $\gamma \in \Gamma$ of $T$
with fixed-point sets of infinite diameter
(\emph{Exemple} 4.2 of \cite{Stal--06}).
\par
(iv)
Let $\gamma$ be an elliptic automorphism of a tree $T$
such that $(\partial T)^\gamma \ne \emptyset$.
Consider a ray with vertex set $(x_n)_{n \ge 0}$
of which the origin $x_0$ is fixed by $\gamma$
and which represents a boundary point fixed by $\gamma$;
then each of the vertices $x_n$ is fixed by $\gamma$.
\par
(v)
Elliptic automorphisms need not be slender.
\end{Rem}

For \emph{(v)}, consider for example the regular tree $T$ of degree $3$,
a vertex $x_0$ of $T$, 
and the three isomorphic connected components $T_1$, $T_2$, $T_3$
obtained from $T$ by deleting $x_0$ and the incident edges.
An appropriate transposition of $T_1$ and $T_2$ 
fixing $T_3$ is not slender,
since $(\partial T)^{\sigma} = \partial T_3$ is open non-empty, 
but a cyclic permutation $\gamma$ of $T_1$, $T_2$, $T_3$
of order~$3$ is slender since $(\partial T)^{\gamma} = \emptyset$.

\vskip.2cm

Let $\Gamma$ be a group which acts by automorphisms on a tree $T$,
and therefore by homeomorphisms on $\partial T$.
Two elements in $\Gamma$ which are hyperbolic
(namely which  act on $T$ by hyperbolic automorphisms) are
\begin{itemize}
\item[$\bullet$]
\emph{transverse} if they do not have any common fixed point in $\partial T$,
equivalently if the intersections of their axis is finite (possibly empty).
\end{itemize}
The action  is 
\begin{itemize}
\item[$\bullet$]
\emph{strongly hyperbolic} if
$\Gamma$ contains a pair of transverse hyperbolic elements.
\end{itemize}
If the action is strongly hyperbolic, 
the next lemma shows that
there is an abundance of transverse pairs.

\begin{Lemma}
% 14
\label{defsh}
Let $\Gamma$ be a group acting strongly hyperbolically 
on a tree~$T$.
\par

For any hyperbolic element $\gamma_0 \in \Gamma$,
there exist infinitely many hyperbolic elements $\gamma_n$, $n \ge 1$,
such that the $\gamma_n$, $n \ge 0$, are pairwise transverse.
\end{Lemma}

\noindent 
\emph{Proof.}~Let $\alpha, \beta \in \Gamma$ be hyperbolic and transverse.
Upon replacing $\alpha$ with $\beta^k \alpha \beta^{-k}$ for an appropriate $k$,
we can assume furthermore that $\gamma_0$ and $\alpha$ are transverse.
\par
Set $\delta_m = \alpha^m \gamma_0 \alpha^{-m}$ for all $m \ge 1$.
Then there exists a subsequence $(\gamma_n)_{n \ge 1}$
of  $(\delta_m)_{m \ge 1}$ such that the $\gamma_n$, $n \ge 0$,
are pairwise transverse.
\hfill $\square$

\vskip.2cm

The next proposition is a restatement of  well-known facts from the literature.
See for example Proposition 2 of \cite{PaVa--91}
and Proposition 7.2 of \cite{Bass--93}.

\begin{Prop}
% 15
\label{transverseh}
Let $T$ be an infinite tree which is not a linear tree.
Let $\Gamma$ be a group which acts on $T$, minimally 
and in such a way that its action on $\partial T$ has no fixed points.
\par

Then the action of $\Gamma$ on $T$ is strongly hyperbolic.
\end{Prop}

\noindent 
\emph{Proof.}~By minimality, $\Gamma \ne \{1\}$ fixes
no vertex and no pair of adjacent vertices.
By Proposition 3.4 of \cite{Tits--70},
this implies that $\Gamma$ contains at least
one hyperbolic element, say $\gamma_1$,
and that $\partial T \ne \emptyset$.
Denote by $\alpha_1 \in \partial T$ the source of $\gamma_1$
and by $\omega_1$ its sink.

By minimality and
by the proof of Corollary 3.5 of \cite{Tits--70},
$T$ is the union of the axis 
of the hyperbolic elements of $\Gamma$.
As $T$ is infinite and is not a linear tree,
it follows that $\partial T$ is infinite
and that there exists an hyperbolic element 
$\gamma_2 \in \Gamma$,
say with source $\alpha_2$ and sink $\omega_2$,
such that 
$\{\alpha_2,\omega_2\} \ne
\{\alpha_1,\omega_1\}$.
Upon replacing $\gamma_2$ by $\gamma_2^{-1}$,
we may assume that 
$\alpha_2 \notin \{\alpha_1,\omega_1\}$.
If $\omega_2 \notin \{\alpha_1,\omega_1\}$,
there is nothing left to prove; 
upon replacing $\gamma_1$ by $\gamma_1^{-1}$ if necessary,
we can assume from now that $\omega_2 = \omega_1$,
and we denote this point by $\omega$.
\par

Since $\Gamma$ does not fix any boundary point in $\partial T$, 
there exists $\delta \in \Gamma$ 
such that $\delta(\omega) \ne \omega$.
Upon exchanging $\gamma_1$ and $\gamma_2$,
we can assume that $\delta(\alpha_1) \ne \omega$.
\par

We claim that all the orbits of $\Gamma$ on $\partial T$ are infinite.
Since any orbit of $\gamma_1^{\mathbf Z}$ 
on $\partial T \smallsetminus \{\alpha_1,\omega\}$ is infinite,
and similarly for $\gamma_2^{\mathbf Z}$
on $\partial T \smallsetminus \{\alpha_2,\omega\}$, 
the only point we have to check is that 
the orbit $\Gamma(\omega)$ is infinite.
Observe that $\gamma_3 \Doteq \delta \gamma_1 \delta^{-1}$
is hyperbolic, that $\alpha(\gamma_3) = \delta(\alpha_1)$,
and that $\omega(\gamma_3) = \delta(\omega)$.
Since $\omega \notin \{\alpha(\gamma_3),\omega(\gamma_3)\}$,
the $\gamma_3^{\mathbf Z}$-orbit of $\omega$ is infinite.
\emph{A fortiori}, the $\Gamma$-orbit of $\omega$ is infinite.
\par

It is a general fact that, if a group $\Gamma$
acts on a set $\Omega$ in such a way that all its orbits are infinite,
and if $F$ is any finite subset of $\Omega$,
there exists $\gamma \in \Gamma$ such that $F$ and $\gamma(F)$
are disjoint; see\footnote{
This is a straightforward consequence of the following lemma
of B.H.~Neumann \cite{NeBH--54}:
a group cannot be covered by finitely many cosets
with respect to subgroups of infinite index.
Other Proof: Theorem 6.6 in \cite{BMMN--98}.
} 
Lemma 2.3 in \cite{NePM--76}.
In our case, it implies that we can choose $\gamma \in \Gamma$ such that
$\gamma(\{\alpha_1,\omega\}) \cap \{\alpha_1,\omega\} = \emptyset$,
namely such that the hyperbolic elements 
$\gamma_1$ and $\gamma \gamma_1 \gamma^{-1}$ are transverse.
\hfill $\square$


\vskip.2cm


Recall that an action of a group $\Gamma$ 
on a topological space $\Omega$ is 
\begin{itemize}
\item[$\bullet$]
\emph{minimal}
if the only $\Gamma$-invariant closed subpaces
of $\Omega$ are $\Omega$ itself and the empty subspace.
\end{itemize}
If a group $\Gamma$ acts on a tree $T$ in a minimal way
(definition before Proposition \ref{Powerstree}),
its action on $\partial T$ \emph{need not be minimal}.
A first example is that of the standard action of $\mathbf Z$ on a linear tree;
a second example is the action of a Baumslag-Solitar group $BS(1,n)$
on its Bass-Serre tree $T$, which  is minimal,
but with the corresponding action on the infinite boundary $\partial T$
having a fixed point (see  Subsection \ref{ssHNN} below).
However:


\begin{Prop}
% 16
\label{minimal}
Let $\Gamma$ be a group 
which acts  on a tree~$T$.
Assume that the action is  strongly hyperbolic  and minimal.
\par
Then the action of $\Gamma$ on  $\partial T$ is minimal.
\end{Prop}

\noindent 
\emph{Proof.}~Let $T_0$ be the subtree of $T$
which is the union of the axis $T_{\gamma}$
over all hyperbolic elements $\gamma \in \Gamma$.
As already noted in the proof 
of Proposition \ref{transverseh}, % 15
$T_0 = T$. Hence the set
\begin{equation*}
L_{\Gamma} \,  \Doteq \, 
\{ \eta \in \partial T \hskip.1cm \vert \hskip.1cm
\eta = \omega(\gamma) \hskip.2cm \text{for some hyperbolic} \hskip.2cm
\gamma \in \Gamma \} 
\end{equation*}
is dense in $\partial T$.
\par

Let $C$ be a non-empty  $\Gamma$-invariant closed
subset of $\partial T$, and choose $\xi \in C$.
Let $\eta \in L_{\Gamma}$; 
choose an hyperbolic element $\gamma \in \Gamma$ 
such that $\eta = \omega(\gamma)$.
In case  $\xi \ne \alpha(\gamma)$,
we have $\eta = \lim_{n \to \infty} \gamma^n (\xi)$,
so that $\eta \in C$.
\par

Assume now that $\xi = \alpha(\gamma)$.
By Lemma \ref{defsh},
there exists an hyperbolic element $\gamma' \in \Gamma$
transverse to $\gamma$.
We have first $\omega(\gamma') \in C$ as in the previous case,
and then  
$\eta = \lim_{n \to \infty} \gamma^n \big( \omega(\gamma') \big) \in C$.
\par

Hence $C \supset L_{\Gamma}$, 
and $C = \overline{L_{\Gamma}} = \partial T$.
In other terms,
the action of $\Gamma$ on $\partial T$ is minimal.
\hfill $\square$

\vskip.2cm

\emph{Remark.}
Proposition \ref{minimal} % 16
shows that, under appropriate hypothesis,
minimality on $T$ implies minimality on $\partial T$.
Here is a kind of converse:
\par

\emph{
Let $T$ be an infinite tree without vertices of degree one.
If a group $\Gamma$ acts on $T$ in such a way that
its action on $\partial T$ is minimal, 
and if $\Gamma$ contains at least one hyperbolic element,
then the action on $T$ is also minimal.}
\par
Indeed, suppose that the action of $\Gamma$ on $T$ is not minimal,
so that $T_0$ defined as above is a non-empty proper subtree of $T$.
Since $T$ does not have any vertex of degree $1$,
there exists a ray in $T$, disjoint from $T_0$, 
which defines a point $\xi \in \partial T$;
denote by $\left( x_n \right)_{n \ge 0}$ the vertices of this ray,
and by $e$ the first edge of this ray,
with $s(e) = x_0$ and $t(e) = x_1$.
Then the shadow $(\partial T)_e$ is a neighbourhood of $\xi$ in $\partial T$,
disjoint from the $\Gamma$-invariant closed subset $\partial T_0$.
Hence the action of $\Gamma$ on $\partial T$ is not minimal.

\vskip.2cm

A homeomorphism $\gamma$ of a Hausdorff space $\Omega$ is
\begin{itemize}
\item[$\bullet$]
\emph{hyperbolic} if it has the following property:
there are two  points  $\alpha, \omega \in \Omega$
fixed by $\gamma$ such that,
for any neighbourhoods $U$ of $\alpha$ and $V$ of $\omega$,
we have $\gamma^n(\Omega \smallsetminus U) \subset V$
and $\gamma^{-n}(\Omega \smallsetminus V) \subset U$
for $n$ large enough.
\end{itemize}
An action of a group $\Gamma$ on $\Omega$ is
\begin{itemize}
\item[$\bullet$]
\emph{strongly hyperbolic}
if $\Gamma$ contains two hyperbolic homeomorphisms
which are \emph{transverse}, namely
without common fixed point.
\end{itemize}
These generalise notions defined above in the following sense:
an hyperbolic automorphism of a tree $T$
induces an hyperbolic homeomorphism of the boundary $\partial T$,
and a group acting on $T$  strongly hyperbolically
is strongly hyperbolic on $\partial T$.
The following proposition is  a reformulation
of Proposition~11 
and Theorem~13 in \cite{Harp--07},
indeed a reformulation of part of \cite{Powe--75}
(see also Proposition \ref{11deHarp--07} below).


\begin{Prop}
% 17
\label{Powersgen}
Let $\Gamma$ be a group which acts by homeomorphisms
on a Hausdorff topological space $\Omega$.~Assume that the action is 
strongly faithful, strongly hyperbolic, and minimal.

Then $\Gamma$ is a Powers group.
\end{Prop}

\noindent 
\emph{Proof.}~
Here is first a general observation.
Let $\Gamma$ be a group which acts by homeomorphisms
on a Hausdorff topological space $\Omega$
containing a $\Gamma$-invariant dense subspace $\Omega_0$.
Then the action of $\Gamma$ on $\Omega$ is strongly faithful
if and only if the action of $\Gamma$ on $\Omega_0$
is strongly faithful.
\par

Set 
$L_{\Gamma} \Doteq \{ \eta \in \Omega
\hskip.1cm \vert \hskip.1cm
\eta = \omega(\gamma) 
\hskip.2cm \text{for some hyperbolic} \hskip.2cm
\gamma \in \Gamma \}$,
as in the proof of Proposition \ref{minimal}. % 16
If the action of $\Gamma$ on $\Omega$ is strongly hyperbolic,
$L_{\Gamma}$ is non-empty, indeed infinite.
If this action is minimal, $L_{\Gamma}$ is dense in $\Omega$.
It follows that the action of $\Gamma$ on $L_{\Gamma}$
is strongly faithful,
so that the hypothesis of Proposition 11 of  \cite{Harp--07}
(= Proposition  \ref{11deHarp--07} here)
are fulfilled.
\hfill $\square$


\vskip.2cm

The particular case of $\Omega = \partial T$ provides
the following variation
on Proposition  \ref{Powerstree}. % 1

\begin{Prop}
% 18
\label{Powerspart}
Let $\Gamma$ be a countable group which acts on a tree $T$. 
Assume that the action is
slender, strongly hyperbolic,  and minimal.
\par
Then $\Gamma$
is a strongly Powers group.
\end{Prop}

\noindent \emph{Proof.}
\emph{First step: Proposition \ref{Powersgen} % 17
applies to $\Gamma$.}
%
The action of $\Gamma$ on $\partial T$ is
strongly faithful by Corollary \ref{slender} % 12 
and minimal by Proposition \ref{minimal}. % 16
It follows from Proposition \ref{Powersgen}  %  17
that $\Gamma$ is a Powers group.
\par

\emph{Second step:  Proposition \ref{Powersgen} % 17
applies to any normal subgroup $N \ne \{1\}$ of $\Gamma$.}
%
The subgroup $N$ has no fixed vertex in $T$.
Otherwise, since the set $V(T)^N$ is $\Gamma$-invariant,
it would coincide with $V(T)$ by $\Gamma$-minimality, 
and $\Gamma$ could not be faithful on $T$.
The same argument shows that $N$ has 
no fixed pair of adjacent vertices in $T$
and (because $\Gamma$ is faithful and minimal on $\partial T$)
no fixed boundary point in $\partial T$.
It follows that $N$ contains hyperbolic elements,
by Proposition 3.4 of \cite{Tits--70}.
\par
The union of the axis of the hyperbolic elements in $N$
is a $\Gamma$-invariant subtree,
and coincides with $T$ by $\Gamma$-minimality.
It follows  that the action of $N$ on $T$ is minimal,
and also strongly hyperbolic
(for example by Proposition \ref{transverseh}). % 15
Proposition \ref{Powersgen} % 17
again implies that $N$ is a Powers group.
\par

\emph{Coda.}
If $N \ne \{1\}$ is subnormal in $\Gamma$,
namely if there exists a chain of subgroups
$N_0 = N \le N_1 \le \cdots \le N_k = \Gamma$
with $N_{j-1}$ normal in $N_j$ for $j = 1, \hdots, k$,
an induction on $j$ based on the second step
shows that $N$ is a Powers group.
\hfill $\square$

\vskip.2cm

\noindent \textbf{Proof of Proposition \ref{Powerstree}.}~  % 1
Since there are in $T$ neither vertices of degree one
nor pending rays,
$\partial T$ is non-empty and perfect.
The action of $\Gamma$ on  $\partial T$  is 
strongly hyperbolic by Proposition \ref{transverseh}, % 15
so that Proposition \ref{Powerspart} applies. % 18
\hfill $\square$

\section{On the action of the fundamental group 
of a graph of groups on the corresponding Bass-Serre tree
}
% 3
\label{section3}

Recall that a \emph{graph of groups} 
$\mathbf G = (G,Y)$ consists of
\begin{itemize}
\item[$\bullet$]
a non-empty connected graph $Y$,
\item[$\bullet$]
two families of groups $\left( G_y \right)_{y \in V(Y)}$
and $\left( G_e \right)_{e \in E(Y)}$,
with $G_{\overline{e}} = G_e$ for all $e \in E(Y)$,
\item[$\bullet$]
a family of monomorphisms $\varphi_e : G_e \longrightarrow G_{t(e)}$,
for $e \in E(Y)$.
\end{itemize}
An \emph{orientation} of $Y$ is a subset  $E^+(Y)$ of $E(Y)$
containing exactly one of $e,\overline{e}$ for each $e \in E(Y)$;
we denote by $E^-(Y)$ the complement of $E^+(Y)$ in $E(Y)$.
\par

A graph of groups $\mathbf G = (G,Y)$ gives rise to
the \emph{fundamental group} $\Gamma = \pi_1(G,Y,M)$ of $\mathbf G$
and the \emph{universal cover} $T =  T (G, Y, M, E^+(Y))$,
also called the \emph{Bass-Serre tree of $\mathbf G$},
where $M$ is a maximal tree in $Y$,
and  $E^+(Y)$ an orientation of $Y$;
abusively, $T$ is also called the
\emph{Bass-Serre tree of $\Gamma$}.
Let us recall as follows  part of the standard theory
($\S$~I.5 in \cite{Serr--77}, % Page 61.
and \cite{Bass--93}).

\begin{itemize}

\vskip.2cm
\item[$\bullet$]
(BS-1)
$\Gamma$ has a presentation with
generators the groups $G_y$,  $y \in V(Y)$,
and elements $\tau_e$,  $e \in E(Y)$, 
and relations
\begin{equation*}
\aligned
\tau_{\overline{e}} \, = \, (\tau_e)^{-1} \hskip1cm 
& \text{for all} \hskip.2cm e \in E(Y) ,
\\
\tau_e^{-1} \varphi_{\overline{e}}(h) \tau_e \, = \, \varphi_{e}(h) \hskip1cm 
& \text{for all} \hskip.2cm e \in E(Y) 
    \hskip.2cm \text{and} \hskip.2cm h \in G_e ,
\\
\tau_e \, = \, 1 \hskip1cm
& \text{for all} \hskip.2cm e \in E(M).
\endaligned
\end{equation*}
\par

Moreover, 
the natural homomorphisms 
\begin{equation*}
G_y \longrightarrow \Gamma
\hskip.5cm \text{and} \hskip.5cm
\mathbf Z \longrightarrow \Gamma , \hskip.2cm k \longmapsto \tau_e^k
\end{equation*}
\emph{are injective} for all $y \in V(Y)$ 
and for all $e \in E(Y)$ with  $e \notin E(M)$.

\vskip.2cm
\item[$\bullet$]
(BS-2)
$T$ is a graph with 
\begin{equation*}
V(T) \, = \,  \bigsqcup_{y \in V(Y)} \Gamma / G_y
\hskip.5cm \text{and} \hskip.5cm
E(T) \, = \,  \bigsqcup_{e \in E(Y)} \Gamma / \varphi_e(G_e) .
\end{equation*}


The source map, the terminus map, and the inversion map,
are given by
\begin{equation*}
\aligned
s(\gamma \varphi_e(G_e)) \, &= \, \left\{
\aligned
\gamma G_{s(e)} \hskip.2cm
& \text{if} \hskip.2cm e \in E^+(Y) 
\\
\gamma {\tau_e}^{-1} G_{s(e)} \hskip.2cm 
& \text{if} \hskip.2cm e \notin E^+(Y) ,
\endaligned
\right.
\\
%
t(\gamma \varphi_e(G_e)) \, &= \, \left\{
\aligned
\gamma \tau_e G_{t(e)} \hskip.7cm
& \text{if} \hskip.2cm e \in E^+(Y) 
\\
\gamma G_{t(e)} \hskip.7cm 
& \text{if} \hskip.2cm e \notin E^+(Y) ,
\endaligned \right.
\\
%
\overline{\gamma \varphi_e(G_e)} \, &= \,  
\gamma \varphi_{\overline e}(G_{\overline{e}})
\endaligned 
\end{equation*}
for all $\gamma \in \Gamma$ and $e \in E(Y)$.
The natural action of $\Gamma$ is by automorphisms of graphs, 
and without inversions.
\par

Moreover, $T$ \emph{is a tree.}

\vskip.2cm
\item[$\bullet$]
(BS-3)
The natural mappings
\begin{equation*}
\aligned
V(T) \longrightarrow V(Y) ,
\hskip.5 cm 
&  \gamma G_y \longmapsto y
\\
E(T) \longrightarrow E(Y) ,
\hskip.5cm
& \gamma \varphi_e(G_e) \longmapsto  e
\endaligned
\end{equation*}
are the constituants of a morphism of graphs 
$p : T \longrightarrow Y$
which factors as an isomorphism 
$\Gamma \backslash T \approx Y$.
\par

\vskip.2cm
\item[$\bullet$]
(BS-4)
The sections
\begin{equation*}
\aligned
V(Y) \longrightarrow V(T) ,
\hskip.5 cm 
&  y \longmapsto \widetilde y \Doteq 1G_y
\\
E(Y) \longrightarrow E(T) ,
\hskip.5cm
& e \longmapsto \widetilde e \Doteq 1 \varphi_e(G_e)
\endaligned
\end{equation*}
are such that the stabilizer of $\widetilde y$ 
in $\Gamma$ is isomorphic to $G_y$
for all $y \in V(Y)$, and similarly the stabilizer of $\widetilde e$ in $\Gamma$
is isomorphic to $G_e$ for all $e \in E(Y)$.

The first section $V(M) = V(Y) \longrightarrow V(T)$
and the restriction $E(M) \longrightarrow E(T)$ of the second section
are the constituents of an isomorphism of graph
from $M$ onto a subtree of $T$.

\end{itemize}
%
Moreover, up to isomorphisms, $\Gamma$ and $T$ 
do not depend on the choices of $M$ and $E^+(Y)$. 
\par
For what we need below, it is important to observe that 
\begin{center}
\emph{the action of $\Gamma$ on $T$ 
need  be neither faithful nor minimal.}
\end{center}
We will often write $G_e$ instead of $\varphi_e(G_e)$;
this is abusive since, though $G_{\overline e} = G_e$, the cosets
$\Gamma / G_e$ and $\Gamma / G_{\overline e}$
are {\it different sets} of oriented edges, 
see the definitions of $E(T)$ and of the change of orientations in (BS-2).

\subsection{Small Bass-Serre trees,
fixed points, and fixed boundary points
}
% 3.1
\label{ssleavespending}

Let $\mathbf G = (G,Y)$ be a graph of groups.
An edge $e \in E(Y)$ is \emph{trivial}\footnote{Terminological adjustment.
``Trivial'', here and on Page 193 of \cite{ScWa--79},
means the same as ``directed'' on Page 1096 of \cite{Bass--76}.
``Reduced'' here means the same as ``minimal''  in \cite{ScWa--79}.
On page 42 of \cite{Bass--93}, a vertex $y \in V(Y)$ is 
\emph{terminal} 
if there is a unique edge $e \in E(Y)$ with $s(e) = y$,
and if $\varphi_{\overline{e}} : G_e \longrightarrow G_{s(e)}$
is an isomorphism onto; observe then that $e$ is trivial.
If $Y$ is a segment of length $2$ with two end vertices $x,z$,
a middle vertex $y$, the vertex group $G_y$ isomorphic to the two
edge groups, and $G_x, G_z$ large enough,
then $(G,Y)$ is not reduced but does not have any terminal vertex.
The contraction process $\mathbf G \rightsquigarrow \mathbf G / e$
is a particular case of the process described with more details
in Section 2 of \cite{Bass--76}.
}  
if $s(e) \ne t(e)$ 
and if $\varphi_{\overline{e}} : G_e \longrightarrow G_{s(e)}$
is an isomorphism onto.
If $\mathbf G = (G,Y)$ has a trivial edge $e$, 
we can define a new graph $Y/e$
obtained from $Y$ by collapsing $\{e,\overline{e}\}$ to a vertex,
and we can define naturally a new graph of groups
$\mathbf G /e$,
with fundamental group isomorphic to that of $\mathbf G$.
Say that $\mathbf G$ is 
\begin{itemize}
\item[$\bullet$]
\emph{reduced}
if it does not contain any trivial edge.
\end{itemize}
\par



Let  $X$ be a connected subgraph of $Y$.
The corresponding \emph{subgraph of groups} $\mathbf F = (F,X)$
is defined by
$F_x = G_x$ for all $x \in V(X)$,
and $F_e = G_e$ for all $e \in E(X)$,
with the inclusion $F_e \longrightarrow F_{t(e)}$
being precisely $\varphi_e$
for all $e \in E(X)$.
\par

The next proposition  
collects observations from papers by Bass; we choose
a maximal tree $M$ of $Y$ containing a maximal tree $L$ of $X$,
and an orientation $E^+(Y)$  of $Y$ containing an orientation $E^+(X)$ of $X$.

\begin{Prop}
% 19
\label{Bass1.14+2.15+2.4}
Let $\mathbf G = (G,Y)$ be a graph of groups,
and let the notation be as above.
\begin{itemize}
\item[(i)]
If $\mathbf F = (F,X)$ is a subgraph of groups of $\mathbf G$, 
the fundamental group $\pi_1(F,X,L)$ is isomorphic
to a subgroup of $\pi_1(G,Y,M)$
and the  universal cover $T (F,X,L,E^+(X))$
can be identified with a subtree of $T (G,Y,M,E^+(Y))$.
\par
In case $\mathbf G$ is, moreover, finite and reduced,
and if $X$ is a proper subgraph of $Y$,
then $\pi_1(F,X,L)$ is a proper subgroup of $\pi_1(G,Y,M)$.


\item[(ii)]
If $Y$ is finite,
there exist a reduced graph of group $\mathbf H = (H,Z)$
and a \emph{contraction}
$\mathbf G = (G,Y) 
\longrightarrow \mathbf H = (H,Z)$
which induces an isomorphism 
from  the fundamental group of $\mathbf G$ 
onto that of $\mathbf H$.
\end{itemize}
\end{Prop}

\noindent 
\emph{Proof.}~
For (i), see
Items 1.14 and 2.15 in \cite{Bass--93}.
For (ii), see
Proposition 2.4 in \cite{Bass--76}.
\hfill $\square$
\par
[Claim (ii) need not hold when $Y$ is infinite;
see the discussion in Section~7 of \cite{ScWa--79}.]

\vskip.2cm




For the next proposition,
compare for example with Theorem 6.1 in \cite{Bass--76}.


\begin{Prop}
\label{finitelinear}
% 20
Let $\mathbf G = (G,Y)$ be a graph of groups,
with universal cover $T$;
we assume  that $\mathbf G$ is reduced. Then:
\begin{itemize}

\item[(i)]
$T$ is finite
if and only if $T$ is reduced to one vertex, 
if and only if $Y$ is reduced to one vertex.

\item[(ii)]
$T$ has no vertex of degree $1$.

\item[(iii)]
If $T$ is infinite, 
it does not have any vertex fixed by $\Gamma$.

\item[(iv)]
$T$ is a linear tree
if and only if  $Y$ is 
%
\subitem{--}
either a segment
with two vertices and one pair of edges $\{e, \overline{e}\}$,
and  with 
$[G_{s(e)} : \varphi_{\overline{e}}(G_e)] = [G_{t(e)} : \varphi_e(G_e)] = 2$
(case of a degenerate non-trivial amalgam, 
see Subsection \ref{ssamalgamated}),
% 4.1
%
\subitem{--}
or a loop with one vertex $y$ and 
one pair of edges $\{e, \overline{e}\}$,
and with 
$\varphi_e(G_e) = G_y = \varphi_{\overline{e}}(G_e)$
(case of a semi-direct product $G_y \rtimes_{\theta} \mathbf Z$,
see Subsection \ref{ssHNN}). % 4.2

\item[(v)]
$T$ has a pending ray if and only if it is a linear tree.

\end{itemize}
\end{Prop}


\noindent 
\emph{Proof.}~
Let $\mathbf G = (G,Y)$ be a reduced graph of groups.
If $Y$ is not reduced to one vertex,  it contains 
either a segment of length one which is not directed, or a loop.
In both cases, $T$ contains a linear subgraph $X$,
by Proposition \ref{Bass1.14+2.15+2.4},
and $\Gamma$ contains an element $\gamma$
which leaves $X$ invariant and which induces on $X$
an hyperbolic translation.
Moreover, any edge of $T$ is contained
in a linear subgraph.
Claims (i) to (iii)  follow.

Suppose that $Y$ contains an edge $e$ with $s(e) \ne t(e)$;
suppose moreover  either that at least one of
$[G_{s(e)} : \varphi_{\overline{e}}(G_e)]$, $[G_{t(e)} : \varphi_e(G_e)]$ 
is at least $3$,
or that $Y$ contains at least one other edge than $e$ and $\overline{e}$.
Then $T$ has an abundance of vertices of degrees at least $3$,
and therefore $T$ does not have pending rays.
Similarly,
 if $Y$ contains a vertex $y$ with at least two loops incident to $y$, 
 then $T$ has an abundance of vertices of degrees at least $4$
 and  $T$ does not have pending rays.
This shows Claims (iv) and~(v).
\hfill $\square$

\subsection{Faithful actions
}
% 3.2
\label{ssfaithful}

Let
$\mathcal N = \left( N_y \right)_{y \in V(Y)}$ 
be the family of subgroups of the  groups in the family
$\left( G_y \right)_{y \in V(Y)}$
defined as follows:
\begin{itemize}
\item[$\circ$]
$N_y$ is a normal subgroup of $G_y$ for all $y \in Y$;
\item[$\circ$]
the family is \emph{$Y$-invariant}, 
which means that, for all $e \in E(Y)$,
there exists a subgroup $N_e$ of $G_e$, 
with $N_{\overline{e}} = N_e$,
such that $N_{t(e)} = \varphi_e (N_e)$;
\item[$\circ$]
and $\mathcal N$ is maximal with these properties
(in the sense that $N_y$ is maximal in $G_y$ for all $y \in V(Y)$).
\end{itemize}
Note that, for all $y,y' \in V(Y)$, the groups $N_y$ and $N_{y'}$
are isomorphic.
\par
Our next proposition is Proposition 1.23 in \cite{Bass--93}.

\begin{Prop}
\label{faithful}
% 21
Let $\mathbf G = (G,Y)$ be a graph of groups
and let $\Gamma, T, \mathcal N = \left( N_y \right)_{y \in V(Y)}$ 
be as above.
Choose a vertex $y_0 \in V(Y)$.
\par
Then the kernel of the action of $\Gamma$ on $T$
is isomorphic to $N_{y_0}$.
In particular, the action of $\Gamma$ on $T$
is faithful if and only if $N_{y_0} = \{1\}$.
\end{Prop}

\subsection{Minimal actions
}
% 3.3
\label{ssminimal}

The next proposition follows from the proof of Corollary 3.5 in \cite{Tits--70},
and is also part of Proposition 7.12 in \cite{Bass--93}.


\begin{Prop}
\label{minimalG}
% 22
Let $\mathbf G = (G,Y)$ be a graph of groups,
and let $\Gamma, T$ be as above;
we assume for simplicity that 
the diameter of the underlying graph $Y$ is finite.
\par

The action of $\Gamma$ on $T$ is minimal
if and only if $Y$ is reduced.
\end{Prop}



\section{The two standard examples
}
% 4
\label{section4}

\subsection{Amalgamated free products
}
% 4.1
\label{ssamalgamated}


In this  case,
the underlying graph $Y$ is a segment  of length one,
with two vertices $x,y$ 
and one pair of edges $\{e,\overline{e}\}$.
The edge group $G_e$ can be identified 
to a subgroup of both $G_x$ and $G_y$.
The fundamental group of $\mathbf G = (G_x, G_y, G_e, Y)$ 
is the free product with amalgamation, 
or, for short, the \emph{amalgam}
$
\Gamma \, = \, G_x \ast_{G_e} G_y .
$
From now on in this subsection, 
we write $A, B, C$  instead of $G_x, G_y, G_e$, 
so that in particular
\begin{equation*}
\Gamma \, = \, A \ast_C B 
\hskip.5cm \text{acts on its Bass-Serre tree} \hskip.2cm T .
\end{equation*}
\par


The edge set of the universal cover $T$ of $\mathbf G$
consists of two copies of $\Gamma / C$
exchanged by the involution
$e \longmapsto \overline{e}$,
say $E(T) = E^+(T) \sqcup E^-(T)$ 
with $E^+(T) = \Gamma / C$ and $E^-(T) = \overline{\Gamma / C}$.
The vertex set of $T$
is the disjoint union $\Gamma / A \sqcup \Gamma / B$.
The source and terminus mappings
are defined to be the canonical projections
\begin{equation*}
\aligned
s : \Gamma / C \longrightarrow \Gamma / A,
\hskip.5cm &\gamma C \longmapsto \gamma A
\\
t : \Gamma / C \longrightarrow \Gamma / B,
\hskip.5cm &\gamma C \longmapsto \gamma B
\endaligned
\end{equation*}
and
\begin{equation*}
\aligned
s : \overline{\Gamma / C} \longrightarrow \Gamma / B,
\hskip.5cm &\overline{\gamma C} \longmapsto \gamma B
\\
t : \overline{\Gamma / C} \longrightarrow \Gamma / A,
\hskip.5cm &\overline{\gamma C} \longmapsto \gamma A .
\endaligned
\end{equation*}
In particular, the tree $T$ is bipartite regular,
with one class of vertices of degree  $[A : C]$
and the other class of degree $[B : C]$.
The action of $\Gamma$ 
has two orbits on the vertex set;
it is transitive on the orientation $E^+(T)$,
or equivalently on the set of geometric edges of $T$.
\par
    

The \emph{kernel} of such an amalgam is the subgroup
\begin{equation}
\label{Eqkeramalgam}
\ker(A \ast_C B)  \, = \, 
\bigcap_{\gamma \in \Gamma} \gamma^{-1} C \gamma 
\end{equation}
of $C$, namely the largest subgroup of $C$
which is normal in both $A$ and $B$.
An amalgam is
\begin{itemize}
\item[$\bullet$]
\emph{faithful} if its kernel is reduced to $\{1\}$,
equivalently if the action of $\Gamma$ on $T$ is faithful.
\end{itemize}
With the notation of Proposition \ref{C*simpleamalg},
observe that
\begin{equation*}
\ker(A \ast_C B)  \, = \, 
\bigcap_{\ell \ge 0} C_\ell \, \subset \, C_k
\hskip.5cm \text{for all} \hskip.2cm k \ge 0 .
\end{equation*}
\par

The amalgam is 
\begin{itemize}
\item[$\bullet$] 
\emph{non-trivial} if $A \ne C \ne B$ 
(equivalently if $\mathbf G$ is reduced), and
\item[$\bullet$]
\emph{non-degenerate} if moreover
at least one of the indices $[A : C]$, $[B : C]$
is strictly larger than $2$.
\end{itemize}
 From the definition of the universal cover $T$ of $\mathbf G$,
the amalgam is trivial if and only if 
the diameter of $T$ is finite
(and this occurs if and only the diameter of $T$ is at most $2$).
Also, the amalgam is non-trivial and degenerate
if and only if $T$ is  a linear tree.

\vskip.2cm

\noindent
\emph{Remarks.}
(i)
A non-trivial amalgam which is faithful is 
\emph{a fortiori} non-degenerate,
unless it is the infinite dihedral group.
\par

(ii)
The condition for $\Gamma$ to act faithfully on $T$
\emph{implies} that $\Gamma$ is icc
(but the converse does not hold); 
see Corollary~2 in \cite{Corn--09}.
Recall that a group $\Gamma$
is \emph{infinite conjugacy class}, or shortly is \emph{icc},
if it is infinite 
and if all its conjugacy classes except $\{1\}$ are infinite.
\par

(iii)
For example, if $m,n \ge 2$ are two coprime integers,
the torus knot group 
$\langle a,b \hskip.1cm \vert \hskip.1cm a^m = b^n \rangle
=
\langle a \rangle \ast_{\langle a^m = b^n \rangle} \langle b \rangle$
\emph{does not} act faithfully on its Bass-Serre tree;
its kernel is infinite cyclic, generated by $a^m = b^n$.
Any non-trivial free product $A \ast B$ acts faithfully on its Bass-Serre tree.
\par

(iv)
Let $T$ be a regular tree of some degree $d \ge 3$.
Let $\Gamma$ denote the group of all automorphisms $\gamma$ of $T$
which are either elliptic or hyperbolic with an even translation length;
it is a subgroup of index $2$ in $\operatorname{Aut}(T)$,
and simple \cite{Tits--70}.
Choose an edge $e \in V(T)$.
Denote by $A$ [respectively $B$, $C$] the pointwise stabiliser in $\Gamma$
of $s(e)$ [respectively of $t(e)$, of $\{s(e),t(e)\}$],
so that $\Gamma = A \ast_C B$.
We leave it to the reader to check that
(as already stated in the introduction), 
for any $k \ge 0$,
the group $C_k$ is the pointwise stabiliser of
the neighbourhood $\mathcal V _k(e)$ defined in the introduction,
and that
\begin{equation*}
\aligned
C \, = \, C_0 \,  \supsetneqq \,   C_1 \,  &\supsetneqq \, \cdots \, 
\supsetneqq  \, C_k  \,  \supsetneqq  \, C_{k+1} \,  \supsetneqq  \, \cdots 
\\
&\supsetneqq  \bigcap_{k \ge 0}C_k 
\, = \, \ker(A \ast_C B) \, = \, \{1\} ,
\endaligned
\end{equation*} 
is a \emph{strictly} decreasing infinite sequence of subgroups of $C$.
Observe that this group $\Gamma$ is not countable,
but that it contains dense countable subgroups
giving rise similarly to strictly decreasing infinite sequences
of $C_k$'s.
We are grateful to Laurent Bartholdi for suggesting these examples.
\par

(v)
For an integer $k \ge 1$, recall that $\Gamma = A \ast_C B$
is \emph{$k$-acylindrical} if, 
whenever $\gamma \in \Gamma$  fixes pointwise
a segment of length $k$ in its Bass-Serre tree,
then $\gamma = 1$ \cite{Sela--97}.
Thus the condition ``$C_k = \{1\}$ for some $k \ge 0$''
is substancially weaker than the condition
``$\Gamma$ is $\ell$-acylindrical for some $\ell \ge 0$''.
The previous remark shows that the condition ``$\ker (A \ast_C B) = \{1\}$''
is the weakest of all.



\begin{Prop}
% 23
\label{PropAmal}
Let $\Gamma = A \ast_C B$ be an amalgam
acting on its Bass-Serre tree $T$ as above.
\begin{itemize}
\item[(i)]
The amalgam is non-trivial if and only if
there is no vertex in $T$ fixed by $\Gamma$,
if and only if the tree $T$ is infinite;
if these conditions hold, then
the action of $\Gamma$ on $T$ is \emph{minimal}.
\end{itemize}
We assume from now on that the amalgam is non-trivial.
\begin{itemize}
\item[(ii)]
The amalgam is non-degenerate if and only if
$T$ is not a linear tree,
if and only if $\partial T$ is perfect,
if and only if
the action of $\Gamma$ on $T$ is \emph{strongly hyperbolic}.
\end{itemize}
We assume from now on that the amalgam is non-degenerate. 
\begin{itemize}
\item[(iii)]
The action of $\Gamma$ on $\partial T$ is minimal.
\item[(iv)]
The action of $\Gamma$ on $T$ is slender
as soon as $C_k = \{1\}$ for some $k \ge 1$.
\end{itemize}
\end{Prop}

\vskip.2cm


\noindent 
\emph{Proof.}~For (i), 
see Propositions \ref{finitelinear} and \ref{minimalG}.
[Alternatively, a direct argument is straightforward.]

\vskip.1cm


(ii) Observe first that, if the amalgam is non-trivial and degenerate,
then $T$ is a linear tree,
so that $\partial T$ has two points, in particular is not perfect,
and the action of $\Gamma$ on $\partial T$
is not strongly hyperbolic.
\par
Suppose now that the amalgam is non-degenerate.
Choose $q,r \in A$ such that the three cosets $C, qC, rC$ in $A/C$
are pairwise disjoint, 
and $s \in B$ with $s \notin C$
(so that $C \cap sC = \emptyset = C \cap s^{-1}C$ in $B/C$).
\par
In $T$, there is a first segment of length $5$, 
of which the $6$ vertices and $5$ oriented edges are,
in ``the'' natural order,
\begin{equation*}
qs^{-1}A, \hskip.2cm  qs^{-1}C, \hskip.2cm
qB, \hskip.2cm qC, \hskip.2cm 
\underline{A}, \hskip.2cm \underline{C}, \hskip.2cm 
\underline{B}, \hskip.2cm \underline{sC}, \hskip.2cm 
\underline{sA}, \hskip.2cm sqC , \hskip.2cm 
sqB,
\end{equation*}
and similarly a second segment of length $5$
with vertices and edges
\begin{equation*}
rs^{-1}A, \hskip.2cm  rs^{-1}C, \hskip.2cm
rB, \hskip.2cm rC, \hskip.2cm 
\underline{A}, \hskip.2cm \underline{C}, \hskip.2cm 
\underline{B}, \hskip.2cm \underline{sC}, \hskip.2cm 
\underline{sA}, \hskip.2cm srC , \hskip.2cm 
srB.
\end{equation*}
These two segments of length $5$
have a common subsegment of length $2$
with vertices and edges underlined above.
\par

The element $sqsq^{-1} \in \Gamma$ maps 
the first edge of the first segment onto its last edge,
and $srsr^{-1} \in \Gamma$ maps
the first edge of the second segment onto its last edge.
It follows that $sqsq^{-1}$ and $srsr^{-1}$
are hyperbolic elements, with axis sharing exactly 
two geometric edges
(see if necessary Proposition 25 in Subsection I.6.4 of \cite{Serr--77}).
Thus these two hyperbolic elements are transverse.
\par

If the amalgam is non-degenerate,
at least every other vertex in $T$ is of degree at least $3$.
In particular, $T$ does not have any pending ray
(Proposition \ref{finitelinear}), so that $\partial T$ is perfect.

\vskip.1cm

(iii)
If the amalgam is non-degenerate,
the action of $\Gamma$ on $T$ is minimal, by (i).
Hence the action is also minimal on $\partial T$, 
by Proposition \ref{minimal}. % 16
\par

\vskip.1cm

(iv)
Let $\gamma \in \Gamma$ be such that 
the fixed point set $(\partial T)^{\gamma}$ of $\gamma$ 
on the boundary has non-empty interior.
To finish the proof, 
it suffices to show that, 
if the amalgam is non-degenerate and if $C_k = \{1\}$,
then $\gamma = 1$.
\par

Since $\partial T$ is perfect, $\gamma$ cannot be hyperbolic.
We can therefore assume that $\gamma$ has a fixed vertex $x_0 \in V(T)$.
Let $(x_n)_{n \ge 0}$ be the vertices of a ray
starting from $x_0$ and representing a boundary point $\xi$
in the interior of $(\partial T)^{\gamma}$.
By Remark  \ref{Remarks}.iv, % 13
the vertex $x_n$ is fixed by $\gamma$ for all $n \ge 0$.
Since $\xi$ is in the interior of $(\partial T)^{\gamma}$,
there is an edge  in the ray  $(x_n)_{n \ge 0}$,
say $d$ from $x_m$ to $x_{m+1}$,
such that $(\partial T)_d \subset (\partial T)^{\gamma}$.
\par

Let $U$ be the subtree of $T$ of which the vertices
belong to rays with first two vertices $x_m, x_{m+1}$.
Since rays in this tree represent 
boundary points in $(\partial T)^{\gamma}$,
the same remark as above implies that
$\gamma$ fixes all vertices and all edges in $U$.
Choose an edge $e \in E(U)$ such that 
all vertices at distance at most $k$ from $s(e)$ or $t(e)$
are in $V(U)$; these vertices are  fixed by $\gamma$.
Choose moreover $\delta \in \Gamma$ such that $e = \delta C$;
we can assume that $s(e) \in \Gamma / A$ and $t(e) \in \Gamma / B$.
\par

Let us first assume for simplicity that $k = 1$.
Choose  transversals $R \subset A$ and $S \subset B$ such that 
$A = \bigsqcup_{r \in R} rC$ and $B = \bigsqcup_{s \in S} sC$
(disjoint unions).
Since $\gamma$ fixes the edges $\delta rC$ and $\delta sC$,
we have $\gamma \in \delta rC(\delta r)^{-1}$ 
and $\gamma \in \delta sC(\delta s)^{-1}$ 
for all $r \in R$ and $s \in S$,
namely
\begin{equation*}
\aligned
\gamma \, &\in \,
\big( \bigcap_{r \in R} \delta rCr^{-1}\delta^{-1} \big)  
\cap
\big( \bigcap_{s \in R} \delta sCs^{-1}\delta ^{-1} \big)
\\
\, &= \,
\delta \left( 
\big( \bigcap_{a \in A} aCa^{-1}  \big)
\cap
\big( \bigcap_{b \in B} bCb^{-1} \big)
\right) \delta ^{-1} 
\\
\, &= \,
\delta C_1 \delta^{-1}
\, = \,
\{1\} ,
\endaligned
\end{equation*}
hence $\gamma = 1$.
\par

The argument in the general case, $k \ge 1$, is similar,
and is left to the reader. Hence, in all cases, $\gamma = 1$.
\hfill $\square$

\vskip.2cm


\noindent
\textbf{Proof of Proposition \ref{C*simpleamalg}.} % 2
We have now to assume that
\begin{center}
$A$, $B$, and therefore also 
$C$, $\Gamma$, and $T$, are countable,
\end{center}
because our proof of Proposition \ref{Baire} % 11
assumes countability. 
Moreover, we assume as in Proposition \ref{C*simpleamalg} % 2
that the amalgam is non-degenerate 
and that $C_k = \{1\}$.

\par
Proposition \ref{PropAmal} % 23
shows that the hypothesis of Proposition \ref{Powerspart} % 18
are satisfied.
Hence the proof of Proposition \ref{C*simpleamalg} is complete.
\hfill $\square$



\subsection{HNN extensions
}
% 4.2
\label{ssHNN}

In this  case,
the underlying graph $Y$ is a loop,
with one vertex $y$, 
and one pair of edges $\{e,\overline{e}\}$.
The edge group $G_e$ can be identified 
(via $\varphi_{\overline{e}}$) to a subgroup of $G_y$,
and we have a monomorphism $\varphi_e : G_e \longrightarrow G_y$.
From now on in this subsection, 
we write $G, H, \theta$ instead of $G_y, G_e, \varphi_{e}$.
The fundamental group of $\mathbf{G} = (G, H, \theta(H), Y)$ is a
\emph{HNN-extension}, which has the presentation
\begin{equation*}
\Gamma \, = \, HNN(G, H, \theta) \, = \,
\langle G, \tau \hskip.1cm \vert \hskip.1cm
\tau^{-1}h\tau = \theta(h) \hskip.2cm
\forall h \in H \rangle 
\end{equation*}
and which acts on its Bass-Serre tree $T$.
\par


The edge set of  $T$ 
consists of two copies of $\Gamma / H$
exchanged by the involution $e \longmapsto \overline{e}$,
say $E(T) = E^+(T) \sqcup E^-(T)$ 
with $E^+(T) = \Gamma / H$ and $E^-(T) = \overline{\Gamma / H}$.
The vertex set of $T$
is  $\Gamma / G$.
The source and terminus mappings
are 
\begin{equation*}
\aligned
s : \Gamma / H \longrightarrow \Gamma / G , \hskip.5cm
&\gamma H \longmapsto \gamma G
\\
t : \Gamma / H \longrightarrow \Gamma / G , \hskip.5cm
&\gamma H \longmapsto \gamma \tau G .
\endaligned
\end{equation*}
and
\begin{equation*}
\aligned
s : \overline{\Gamma / H} \longrightarrow \Gamma / G , \hskip.5cm
&\overline{\gamma H} \longmapsto \gamma \tau G
\\
t : \overline{\Gamma / H} \longrightarrow \Gamma / G , \hskip.5cm
&\overline{\gamma H} \longmapsto \gamma  G .
\endaligned
\end{equation*}
In particular, the tree $T$ is regular, 
of degree $[G:H] + [G : \theta(H)]$.
The action of $\Gamma$ is transitive  
on the vertex set,
as well as on the orientation $E^+(T)$, 
or equivalently on the set of geometric edges of $T$.
\par


The \emph{kernel} of an HNN-extension is the subgroup
\begin{equation}
\label{EqkerHNN}
\ker(HNN(G,H,\theta))  \, \Doteq \, 
\bigcap_{\gamma \in \Gamma} \gamma^{-1} H \gamma 
\end{equation}
of $H \cap \theta(H)$, namely the largest subgroup of $H \cap \theta(H)$
which is both normal in $G$ and invariant by $\theta$.
An HNN extension is
\begin{itemize}
\item[$\bullet$]
\emph{faithful} if its kernel is reduced to $\{1\}$,
equivalently if the action of $\Gamma$ on $T$ is faithful.
\end{itemize}
With the notation of Proposition \ref{C*simpleHNN}, 
observe that
\begin{equation*}
\ker(HNN(G,H,\theta)) \, = \,
\bigcap_{\ell \ge 0} H_{\ell} \, \subset \, H_k
\hskip.5cm \text{for all} \hskip.2cm k \ge 0.
\end{equation*}


\par

The HNN-extension is 
\begin{itemize}
\item[$\bullet$] 
\emph{ascending} 
if at least one of $H, \theta(H)$ is the whole of $G$,
\item[$\bullet$]
\emph{strictly ascending} 
if exactly one of $H, \theta(H)$ is the whole of $G$, and
\item[$\bullet$]
\emph{non-degenerate} if at least one of $H, \theta(H)$ 
is a proper subgroup of $G$.
\end{itemize}
The HNN-extension is degenerate 
if and only if $T$ is a linear tree,
in which case $\theta$ is an automorphism of $G$
and $\Gamma$ is the corresponding semi-direct product
$G \rtimes_{\theta} \mathbf Z$.


\vskip.2cm

\noindent
\emph{Remarks.}
(i)
An HNN-extension with $H \ne \{1\}$
which is faithful is \emph{a fortiori} non-degenerate.
\par

(ii)
The condition for $\Gamma$ to act faithfully on $T$
\emph{implies} that $\Gamma$ is icc
(but the converse does not hold); 
see Example 2.9 in \cite{Stal--06},
and Corollary~4 in \cite{Corn--09}.
\par

(iii)
For example, if $m,n$ are  integers, 
the Baumslag-Solitar group
\begin{equation*}
BS(m,n) \, = \, 
\langle \tau,b \hskip.1cm \vert \hskip.1cm \tau^{-1}b^m\tau = b^n \rangle
\, = \,
HNN(b^{\mathbf Z}, b^{m\mathbf Z}, b^{mk} \longmapsto b^{nk})
\end{equation*}
acts faithfully on the corresponding tree
if and only if $\vert m \vert \ne \vert n \vert$.
Indeed, on the one hand,  if $n = \pm m$, then
$H = \langle b^m \rangle$ is clearly 
a normal subgroup of  $\Gamma$;
and, on the other hand,
it is a result of Moldavanskii
that $BS(m,n)$ has an  infinite cyclic normal subgroup
if and only if $\vert m \vert = \vert n \vert$;
see \cite{Mold--91}, 
or the exposition in the Appendix of \cite{Souc--01}.



\begin{Prop}
% 24
\label{PropHNN}
Let $\Gamma = HNN(G,H,\theta)$ be an HNN-extension
acting on its Bass-Serre tree $T$ as above. Then:
\begin{itemize}
\item[(i)]
There is no vertex in $T$ fixed by $\Gamma$,
the tree $T$ is infinite and the action of $\Gamma$ on $T$ is minimal.
\item[(ii')]
The HNN-extension is non-degenerate if and only if 
$T$ is not a linear tree,
if and only if the space $\partial T$ is perfect.
\end{itemize}
We assume from now on that the extension is non-degenerate.
\begin{itemize}
\item[(ii'')]
The HNN-extension is non-ascending if and only if 
the space $\partial T$ is  without $\Gamma$-fixed point,
if and only if the action of $\Gamma$ on $T$ is strongly hyperbolic.
\end{itemize}
We assume from now on that the extension is non-ascending.
\begin{itemize}
\item[(iii)]
The action of $\Gamma$ on $\partial T$ is minimal.
\item[(iv)]
The action of $\Gamma$ on $T$ is slender
as soon as $H_k = \{1\}$ for some $k \ge 1$.
\end{itemize}
\end{Prop}


\noindent 
\emph{Proof.}~
For (i) and (iii), see the proof of Proposition \ref{PropAmal}. % 23

\vskip.1cm

(ii') If the extension is degenerate, $T$ is a linear tree,
so that $\partial T$ has exactly two points.
Otherwise, $T$ is regular of degree has least $3$,
so that $\partial T$ is perfect.

\vskip.1cm

(ii'')
Suppose first that $H = G \supsetneqq \theta(H)$. The mapping 
$E^+(T) \longrightarrow V(T), \hskip.2cm e \longmapsto s(e)$
is a bijection. 
Choose a geodesic in $T$ with vertex set 
$\left( x_p \right)_{p \in \mathbf Z}$
such that $d(x_p, x_{p'}) = \vert p - p' \vert$
for all $p,p' \in \mathbf Z$,
and such that the edge from $x_p$ to $x_{p+1}$ 
lies in $E^+(T)$ for all $p \in \mathbf Z$;
consider the limit $\xi = \lim_{p \to \infty} x_p \in \partial T$.
Consider also a geodesic ray with vertex set
 $\left( y_q \right)_{q \in \mathbf N}$
such that $d(y_q, y_{q'}) = \vert q - q' \vert$
for all $q,q' \in \mathbf N$,
and such that the edge from $y_q$ to $y_{q+1}$ 
lies in $E^+(T)$ for all $q \in \mathbf N$;
set $\eta = \lim_{q \to \infty} y_q \in \partial T$.
We claim that $\eta = \xi$,
from which it follows that $\xi$ is fixed by $\Gamma$.
\par

To prove the claim, it suffices to check that
there exists $b \in \mathbf Z$ and $q \in \mathbf N$
such that $y_q = x_{q+b}$;
inded, since $E^+(T) \longrightarrow V(T)$, $e \longmapsto s(e)$,
is a bijection, this implies that $y_{q'} = x_{q' + b}$ for all $q' \ge q$,
and therefore $\eta = \xi$.
\par

If one could not find $b,q$ such that $y_q = x_{q+b}$,
there would exist a segment with vertex set
$\left( z_r \right)_{0 \le r \le N}$, with $N \ge 1$,
connecting the geodesic to the ray,
namely with $z_0 = x_{p_0}$ for some $p_0 \in \mathbf Z$
and $z_N = y_{q_0}$ for some $q_0 \in \mathbf N$,
moreover with
$z_r \notin \left\{ x_p \right\}_{p \in \mathbf Z}$ for $r \ge 1$
and 
$z_r \notin \left\{ y_q \right\}_{q \in \mathbf N}$ for $r \le N-1$.
On the one hand, the edge with source $z_{r+1}$ and terminus $z_r$
would be in $E^+(T)$ for all $r$
(as one checks inductively for $r = 0, 1, \hdots$),
on the other hand, the edge with source $z_{r-1}$ and terminus $z_r$
would also be in $E^+(T)$ for all $r$
(as one checks inductively for $r = N, N-1, \hdots$).
But this is impossible, since $E^+(T)$ cannot contain both
some edge $e$ with source $z_{r+1}$ and terminus $z_r$
and the edge $\overline{e}$.
\par

Suppose next that $H \subsetneqq G = \theta(H)$,
so that the mapping 
$E^+(T) \longrightarrow V(T), \hskip.2cm e \longmapsto t(e)$
is a bijection.
An analogous argument shows that there exists a point
in $\partial T$ fixed by $\Gamma$.
\par

It follows also that, if the HNN-extension is ascending,
there cannot exist two transverse hyperbolic elements in $\Gamma$.
\par

Suppose now that the extension is non-ascending.
Choose $r,s \in G$ with $r \notin H$ and $s \notin \theta(H)$.
\par
In $T$, there are two segments of length $2$,
with vertices and edges respectively
\begin{equation*}
\begin{matrix}
\tau^{-1}G & \tau^{-1}H & G & rH & r\tau G ,
\\
s^{-1}\tau^{-1}G & s^{-1}\tau^{-1}H & G & H & \tau G ,
\end{matrix}
\end{equation*}
sharing just one vertex, $G$.
It follows that $r\tau$ and $\tau s$ are two elements of $\Gamma$
which are hyperbolic and transverse,
with axis having in common the unique vertex $G$.

\vskip.1cm

(iv) The proof of this claim is a somewhat tedious variation on that
of Proposition \ref{PropAmal} % 23
and is left to the reader.
\hfill $\square$




\vskip.2cm

\noindent
\textbf{Proof of Proposition \ref{C*simpleHNN}.} % 4
We have now to assume that
\begin{center}
$G$, and therefore also $H$, $\Gamma$, and $T$, are countable,
\end{center}
because our proof of Proposition \ref{Baire} assumes countability. 
Moreover, we assume as in Proposition \ref{C*simpleHNN}
that the HNN-extension is non-ascending and that $H_k = \{1\}$.
\par
Proposition \ref{PropHNN} % 24
shows that the hypothesis of Proposition \ref{Powerspart} % 18
are satisfied.
Hence the proof of Proposition \ref{C*simpleHNN} % 4 
is complete.
\hfill $\square$


\vskip.2cm

\begin{Lemma}
% 25
\label{LemmaBS}
Consider the Baumslag-Solitar group
\begin{equation*}
\Gamma = \langle \tau, b \hskip.1cm \vert \hskip.1cm 
\tau^{-1}b^m\tau = b^n \rangle
\end{equation*}
acting on its Bass-Serre tree $T$. 
Denote by $G$ the cyclic subgroup of $\Gamma$ generated by $b$,
and by $H$ that generated by $b^m$.
Set
\begin{equation*}
K_+ \, = \, \bigcap_{p \ge 0} \tau^{-p} H \tau^{p} 
\hskip.5cm \text{and} \hskip.5cm
K_- \, = \, \bigcap_{p \ge 0} \tau^{p} H \tau^{-p} .
\end{equation*}


(i) We have
\begin{itemize}
\item[$\circ$] 
If $\vert m \vert = \vert n \vert$, then $K_+ = K_- = H$,
\item[$\circ$] 
if $n = \pm am$ for some $a \ge 2$, then $K_+ = \{1\}$ and $K_- = H$,
\item[$\circ$] 
if $m = \pm an$ for some $a \ge 2$, then $K_+ = H$ and $K_- = \{1\}$,
\item[$\circ$] 
In all other cases,  $K_+ = K_- = \{1\}$.
\end{itemize}
\par

(ii) Assume that $\vert m \vert \ne \vert n \vert$.
For any $k \in \mathbf Z$, $k \ne 0$, the automorphism $g = b^k$
of $T$ is elliptic and slender.
\par

(iii) If $\vert m \vert \ne \vert n \vert$, the action of $\Gamma$ on $T$ is slender.
\end{Lemma}

\noindent 
\emph{Proof.}~(i) The normal form theorem for HNN extensions
(Theorem 2.1 of Chapter IV in \cite{LySc--77})
implies that
\begin{equation*}
\aligned
\tau^{-1} b^\ell \tau \, &\notin \, G
\hskip.5cm \text{if} \hskip.2cm \ell \notin m \mathbf Z,
\\
\tau^{-1} b^{km} \tau \, &= \, b^{kn}
\hskip.5cm \text{for all} \hskip.2cm k \in \mathbf Z,
\\
\tau b^\ell \tau^{-1} \, &\notin \, G
\hskip.5cm \text{if} \hskip.2cm \ell \notin n \mathbf Z,
\\
\tau b^{kn} \tau^{-1} \, &= \, b^{km}
\hskip.5cm \text{for all} \hskip.2cm k \in \mathbf Z .
\endaligned
\end{equation*}
Claim (i) follows.

\vskip.1cm

For (ii) and (iii), we assume that $K_- = \{1\}$.
The case with $K_+ = \{1\}$ will follow,
since $HNN(G,H,\theta) \approx HNN(G, \theta(H), \theta^{-1})$.
Let  $x_0$ denote the vertex $G = 1G \in \Gamma / G = V(T)$.
Observe that $x_0$ is fixed by $g$;
indeed, the isotropy subgroup 
$\{\gamma \in \Gamma \hskip.1cm \vert \hskip.1cm \gamma x_0 = x_0 \}$
coincides with $G$.

\vskip.1cm

(ii) 
It suffices to show the following claim:
$(\partial T)_e \not\subset (\partial T)^g$ 
for any $e \in E(T)$ with $d(x_0,s(e)) < d(x_0,t(e))$.
If $g(t(e)) \ne t(e)$, then 
$g\left( (\partial T)_e \right) \cap (\partial T)_e = \emptyset$,
and the claim is obvious; we can therefore assume that $g(t(e)) = t(e)$.
We distinguish two cases, 
depending on the $\Gamma$-orbit of~$e$.
\par

Suppose first that $e \in \Gamma / H$, namely that
there exists $\gamma \in \Gamma$ such that
$s(e) = \gamma \tau^{-1} (x_0)$ and $t(e) = \gamma (x_0)$.
Then $\left( \gamma \tau^p G \right)_{p \in \mathbf N}$
are the vertices of a ray $\rho$ in $T$ starting at $t(e)$,
with $d(\gamma \tau^{p} G, \gamma \tau^{q}G) = \vert p-q \vert$
for all $p,q \in \mathbf N$,
and extending the segment from $x_0$ to $t(e)$. 
For $p \ge 1$, the vertex $\gamma \tau^p G$ is fixed by $g$
if and only if
$g  \in  \gamma G \gamma^{-1} \cap \gamma \tau^p G \tau^{-p} \gamma^{-1}$,
namely if and only if
$\gamma^{-1} g \gamma  \in  G \cap  \tau^p G \tau^{-p}$.
It follows from (i) that there exists an edge $f$ in the ray $\rho$,
with $d(x_0, s(f)) < d(x_0,t(f))$, 
such that $g(s(f)) = s(f)$ and $g(t(f)) \ne t(f)$,
and consequently such that 
$g\left( (\partial T)_f \right) \cap (\partial T)_f = \emptyset$.
Since $(\partial T)_f  \subset (\partial T)_e$,
this implies $(\partial T)_e \not\subset (\partial T)^g$.
\par

Suppose now that $e \in \overline{\Gamma / H}$, namely that
there exists $\gamma \in \Gamma$ such that
$s(e) = \gamma \tau (x_0)$ and $t(e) = \gamma (x_0)$.
Observe that $H \ne G$, otherwise $K_- = G$ could not be $\{1\}$.
Choose $u \in G$ with $u \notin H$,
so that $\overline{\gamma u H}$ is an edge with source
$\gamma u \tau G \ne s(e) = \gamma \tau G$
and with terminus $\gamma G = t(e)$. 
Then $(\gamma u \tau^p G)_{p \in \mathbf N}$
are the vertices of a ray in $T$ 
starting at $\gamma G = \gamma u G = t(e)$,
and the argument of the previous case carries over.

\vskip.1cm

(iii) Let $\gamma \in \Gamma$, $\gamma \ne 1$,
be an elliptic automorphism of $T$.
Choose $\delta \in \Gamma$ such that
the vertex $\delta G$ is fixed by $\gamma$.
Then $g  \Doteq \delta^{-1} \gamma \delta$ fixes $x_0$,
so that $g = b^k$, as in Claim (ii).
We have $(\partial T)^{\gamma} = \delta \left( (\partial T)^g \right)$;
in other words, the subspace $(\partial T)^{\gamma}$ of $\partial T$
is the image by the homeomorphism $\delta$ of $\partial T$
of the subspace  $(\partial T)^g$, without interior points by (ii).
Hence $\gamma$ is slender.
\hfill $\square$

\vskip.2cm 

\noindent
\textbf{Proof of Proposition \ref{BS}.} % 5
If $\min \{ \vert m \vert, \vert n \vert \} = 1$,
the Baumslag-Solitar group $BS(m,n)$ is solvable,
in particular amenable, and therefore it is not C$^*$-simple.
If $\vert m \vert = \vert n \vert$, the group $BS(m,n)$
contains an infinite cyclic normal subgroup
(as already noted in Remark (iii) 
just before Proposition \ref{PropHNN}), % 24
so that $BS(m,n)$ is not C$^*$-simple. 
\par
Let us assume that 
$\min \{ \vert m \vert, \vert n \vert \} \ge 2$
and that $\vert m \vert \ne \vert n \vert$.
Observe that, say in the case of $m$ and $n$ coprime to simplify the discussion,
we have  $G = \mathbf Z$ and
\begin{equation*}
\aligned
H_0 = m \mathbf Z \supsetneqq H_1 = (mn)^2  \mathbf Z \supsetneqq  \cdots
&\supsetneqq  H_k = (mn)^{2k} \mathbf Z \supsetneqq  \cdots 
\\
&\supsetneqq  \bigcap_{k \ge 0}H_k = \ker(BS(m,n)) = \{0\} ,
\endaligned
\end{equation*} 
so that we cannot apply Proposition  \ref{C*simpleHNN}, % 4
even though the action of $BS(m,n)$ on $T$ is faithful
(by this same Remark (iii)).
\par

However,  the action of $BS(m,n)$ on
its Bass-Serre tree $T$ 
is slender, by Lemma \ref{LemmaBS}, %  25
strongly hyperbolic and minimal, by Proposition \ref{PropHNN}. % 24
Hence, the hypothesis of Proposition \ref{Powerspart} % 18
are satisfied,
and the proof of Proposition \ref{BS} % 5 
for $BS(m,n)$ is complete.
\par
The proof of the more general case, which is similar, is left to the reader.
\hfill $\square$



\subsection{About the general case
}
% 4.3
\label{ssremarks1}

Let $\mathbf G = (G,Y)$ be a  graph of groups.
If $Y$ is a tree, the analysis of the fundamental group of $\mathbf G$
can be done essentially as in Subsection \ref{ssamalgamated}.
% 4.1
Assume now that $Y$ is not a tree.
\par

Choose a maximal tree $M$ in $Y$.
Let $Z$ denote the graph obtained from $Y$ 
by collapsing $M$ to a vertex,
namely the wedge of circles with a unique vertex, say $z$,
and as many geometrical loops 
as there are geometrical edges in $Y \smallsetminus M$;
and let $\mathbf H = (H,Z)$ be the corresponding graph of groups,
with $H_z$ the fundamental group 
of the subgraph of groups of $\mathbf G$ defined by $M$,
and with $H_e$ the group $G_e$ for any edge in $E(Y) \smallsetminus E(M)$,
appropriately identified to a subgroup of $H_z$.
Observe that, if $\mathbf G$ is reduced, so is $\mathbf H$.
Then  the fundamental group of $\mathbf H$
is isomorphic to that of $\mathbf G$, 
and a multiple HNN extension of the form
%%%
\begin{equation*}
\Gamma 
\, = \, 
\left\langle H, \tau_e, \hdots, \tau_f \hskip.1cm \Bigg\vert \hskip.1cm
%
\aligned
\tau_e^{-1}h\tau_e &= \theta_e(h) \hskip.2cm
\forall h \in H_e 
\\
& \cdots
\\
\tau_f^{-1}h\tau_f &= \theta_f(h) \hskip.2cm
\forall h \in H_f 
\endaligned
%
\right\rangle ,
\end{equation*}
%%%
where $e, \hdots, f$ are the loops in some orientation
$E^+(Z)$ of the wedge of circles $Z$.
The Bass-Serre tree of $\mathbf H$ is regular of degree
\begin{equation*}
[H_z : H_e] \, + \, [H_z : \theta_e(H_e)] \, + \, \cdots \, + \, 
[H_z : H_f] \, + \, [H_z : \theta_f(H_f)] .
\end{equation*}
The analysis of $\Gamma$ can be done 
essentially as in Subsection \ref{ssHNN}. % 4.2



\subsection{Remark
}
% 4.4
\label{ssremarks2}

On several occasions in the past, 
including in \cite{Harp--85} and \cite{BeHa--86}, 
the first author has been wrong
in dealing with  elliptic automorphisms of a tree $T$,
concerning their fixed point sets in $V(T)$ and in $\partial T$.
As a consequence of this,
Claims (d) and (e) of Theorem 5 in \cite{BeHa--86} 
are not correct as stated, as pointed out in \cite{Stal--06}
(many thanks to Yves Stalder).
The notion of \emph{slender} automorphism
provides some way to fix
at least part of the confusion.



\vskip.4cm
\begin{center}
\textbf{II. Fundamental groups of $3$-manifold}
\end{center}

Manifolds which appear below are assumed to be \emph{connected},
except explicit exceptions.
In Items (5.1) to (5.3), $M$ need not be orientable;
from (5.4) onwards, $M$ is assumed to be orientable.

\section{A reminder on $3$-manifolds and their groups
}
% 5
\label{section5}


Let $M$ be a compact\footnote{This can be slightly relaxed,
because of the following theorem.
\emph{
Let $M$ be a $3$-manifold with $\pi_1(M)$ finitely generated.
Then there exists a compact $3$-submanifold $Q$ of $M$
such that the induced homomorphism
$\pi_1(Q) \longrightarrow \pi_1(M)$ is an isomorphism
}
(Theorem 8.6 in \cite{Hemp--76}).
}
$3$-manifold; set $\Gamma = \pi_1(M)$.
We use $\approx$ to indicate both 
a homeomorphism of manifolds
and an isomorphism of groups.
We also use rather standard notation for particular manifolds,
with the dimension in superscript:
$\mathbf S^n$ for spheres, 
$\mathbf P^n$ for real projective spaces,
$\mathbf I$, $\mathbf D^2$, and $\mathbf B^3$ 
for the interval, the $2$-disc, and the $3$-ball,
$\mathbf T^2$ and $\mathbf {K}^2$ 
for the $2$-torus and the Klein bottle.


\vskip.2cm

(5.1)
The manifold $M$ is \emph{prime}
if it is not homeomorphic to the $3$-sphere and if,
for any connected sum $M_1 \sharp M_2$ homeomorphic to $M$,
one of $M_1$, $M_2$ is a standard $3$-sphere.
Any compact $3$-manifold has  a decomposition in connected sum
$M \approx M_1 \sharp \cdots \sharp M_k$,
with the $M_j$ prime, 
and with a small dose of non-uniqueness 
which can be precisely described;
we have moreover a free product decomposition
$\pi_1(M) = \pi_1(M_1) \ast \cdots \ast \pi_1(M_k)$,
which is unique up to the order of the factors.
This is the \emph{Kneser-Milnor decomposition} of $M$
(Chapter 3 of \cite{Hemp--76}).
\par

Note that we can have $\pi_1(M_j) = \{1\}$ for some $j$.
Standard arguments of algebraic topology show that
this can occur only if
$M_j$ is a $3$-ball or a non-standard $3$-sphere.
Those $M_j$ which are $3$-balls correspond bijectively
to spherical connected components of $\partial M$
(Lemma 3.7 in \cite{Hemp--76}).
Denote by\footnote{Thus
$\mathcal P (M) = \widehat M$
modulo the Poincar\'e Conjecture.
}
\begin{itemize}
\item[$\bullet$]
 $\mathcal P (M) = M_1 \sharp \cdots \sharp M_{\ell}$
the \emph{Poincar\'e completion} of $M$
\end{itemize}
(as in Chapter 10 of \cite{Hemp--76}),
where $M_1, \hdots, M_{\ell}$ stand now 
for the non simply connected prime factors of $M$;
then $\pi_1(\mathcal P (M)) \approx \pi_1(M)$,
and $\pi_1(M_j)$ is not the trivial group for $j = 1, \hdots, \ell$. 
\par

To sum up this part of the discussion, 
there are three cases to distinguish:
\begin{itemize}
\item[(i)] 
$\pi_1(M)$ is a non-degenerate free product,
\item[(ii)]
$\pi_1(M)$ is an infinite dihedral group,
\item[(*)]
$\mathcal P(M)$ is prime. \end{itemize}
(Beware: these cases  overlap, see Remark \ref{Remarks27}.) % 27

\vskip.2cm

(5.2)
A $3$-manifold $M$ is \emph{irreducible} 
if any $2$-sphere in $M$ bounds a $3$-ball.
A prime manifold $M$ is either irreducible
or a $2$-sphere bundle over a circle
(Lemma 3.13 in \cite{Hemp--76}).
Suppose $M$ is a $\mathbf S^2$-bundle over $\mathbf S^1$;
if $M$ is orientable,  this bundle is trivial, 
$M \approx \mathbf S^2 \times \mathbf S^1$;
if $M$ is non-orientable, this bundle is 
the non-trivial $\mathbf S^2$-bundle over $\mathbf S^1$,
say $\mathbf S^2 \widetilde \times \mathbf S^1$;
in both cases, $M$ is a Seifert manifold.
\par

Thus, we can modify the summing up of (5.1),
and replace (*) by (iii) and  (iv):
\begin{itemize}
\item[(iii)]
$\mathcal P (M)$ is a Seifert manifold,
\item[(iv)]
$\mathcal P (M)$ is irreducible and is not a Seifert manifold.
\end{itemize}


\vskip.2cm

(5.3)
A manifold $M$ is \emph{$\partial$-irreducible}
if it is irreducible and if, moreover, 
all the connected components of its boundary 
are incompressible.
(We refer to Chapter 6 of \cite{Hemp--76} for the definition
of ``incompressible''; recall also that the fundamental group
of any incompressible component of the boundary
injects in the fundamental group of $M$.)

\begin{Lemma}
% 26
\label{torusKlein}
Let $M$ be a $3$-manifold such that $\mathcal P (M) = M$.
Assume that $M$ is irreducible and not $\partial$-irreducible.
\par
Then
either $\pi_1(M)$ is a non-trivial free product,
or $M$ is  a solid torus or a solid Klein bottle.
\end{Lemma}

\noindent 
\emph{Proof.}~Since $\partial M$ has a compressible component, say $F$,
there exists a $2$-disc $D$ in $M$ such that
$\partial D = D \cap F$,
with $\partial D$ not contractible in~$F$.
Set $M_1 = M \smallsetminus D$;
more precisely, 
choose an open regular neighborhood 
$\mathcal N(D)$ of $D$ 
and consider the complement $M_1$ of $\mathcal N(D)$ in $M$
(thus $M_1$ is obtained by ``splitting $M$ along $D$'',
and our $M \smallsetminus D$
is denoted by $\sigma_D(M)$ on Page 4 of \cite{JaSh--79}).
There are two cases to distinguish.
\par

Suppose first that  $M_1$ is connected.
By the Seifert -- Van Kampen theorem
(for which the reference we like best is $\S$ 3 of \cite{Rham--69}),
we have $\pi_1(M) = \pi_1(M_1) \ast \mathbf Z$.
If $M_1$ is simply connected,
then $M_1$ is homeomorphic to a $3$-ball
(otherwise $M$ would not be irreducible),
and $M$ is homeomorphic to either a solid torus
or a solid Klein bottle, depending on the action
of the gluing map on the orientation of $D$.
If $M_1$ is not simply connected,
then $\pi_1(M)$ is a non-trivial free product.
\par
If $M_1$ has two connected components,
say $M'$ and $M''$, then $\pi_1(M) = \pi_1(M') \ast \pi_1(M'')$,
again by the Seifert  -- Van Kampen Theorem.
Observe that neither $M'$ nor $M''$ is simply connected
(otherwise one of $M'$, $M''$ would be a $3$-ball,
and $\partial D$ would be contractible in $F$),
so that $\pi_1(M)$ is a non-trivial free product.
\hfill $\square$

\begin{Rem}
% 27
\label{Remarks27}
(i) Each case in the conclusion of Lemma \ref{torusKlein} % 26
occurs.
\par
(ii)
The fundamental group of a $\partial$-irreducible
compact $3$-manifold  is not a non-trivial free product.
\end{Rem}

\noindent \emph{Proof.}
(i) Let $M_1$ be a compact $3$-manifold 
which is irreducible and which has a non-empty boundary.
Let $D,D'$ be two disjoint $2$-discs in $\partial M_1$,
and let $M$ be the result of attaching a handle
$\mathbf D^2 \times \mathbf I$ to $D$ and $D'$.
Then $\pi_1(M) \approx \pi_1(M_1) \ast \mathbf Z$.
The manifold $M$ is irreducible and, 
if  $M_1$ is not simply connected, 
$\pi_1(M)$ is a non-trivial free product; 
observe that $M$ is not $\partial$-irreducible.
\par


(ii) This is a result of Stallings (previously a conjecture of Kneser);
see Theorem 7.1 in \cite{Hemp--76}).
\hfill $\square$

\vskip.2cm

Here is one way to sum up the discussion above.

\begin{Prop}
% 28
\label{3manifoldtrichotomy}
Let $M$ be a compact $3$-manifold. 
Then  at least one of the following statements is true:
\begin{itemize}
\item[(i)]
the group $\pi_1(M)$ is a non-degenerate free product,
and is therefore a strongly Powers group.
\item[(ii)]
$\pi_1(M)$ is an infinite dihedral group, 
\item[(iii)]
$\mathcal P (M)$ is a Seifert manifold,
\item[(iv)]
$\mathcal P (M)$ is $\partial$-irreducible, is not a Seifert manifold,
and $\pi_1(M)$ is not a non-trivial free product.
\end{itemize}
\end{Prop}



(5.4)
Suppose that $\pi_1(M)$ is an infinite dihedral group,
and moreover that $M$ is orientable;
assume also that the Poincar\'e conjecture is true.
Case (ii) in Proposition \ref{3manifoldtrichotomy}
is then contained in Case (iii);
indeed, it follows from a theorem of  \cite{Tao--62} 
that $\widehat M$ is homeomorphic to $\mathbf P^3 \sharp \mathbf P^3$.
Recall that $\mathbf P^3 \sharp \mathbf P^3$
is a circle bundle over $\mathbf P^2$,
and \emph{a fortiori} a Seifert manifold.
\par

If $\pi_1(M)$ is a non-degenerate free product, 
$\mathcal P (M)$ cannot be a Seifert manifold,
because the fundamental group of a Seifert manifold
not covered by $\mathbf S^3$ 
has a normal infinite cyclic subgroup
(Lemma 3.2 in \cite{Scot--83}).
Thus:
\begin{itemize}
\item[$\bullet$]
\emph{for orientable manifolds, exactly one of the
statements (i), (iii), (iv) of Proposition \ref{3manifoldtrichotomy} is true.}
\end{itemize}

\vskip.2cm

(5.5)
An irreducible manifold $M$ is \emph{atoroidal}
if any incompressible torus in $M$ 
is isotopic to a component of $\partial M$.
Here is the basic \emph{JSJ decomposition} theorem 
of Jaco-Shalen and Johansen
(see Page 157 of \cite{JaSh--79},
Page 483 of \cite{Scot--83},
and Theorem 3.4 of \cite{Bona--02}):

\begin{itemize}
\item[$\bullet$]
\emph{Let $M$ be a compact $3$-manifold
which is  $\partial$-irreducible and orientable.
There exists a minimal finite family of disjoint
essential tori $T_1, \hdots, T_k$ such that 
each connected component of 
$M \smallsetminus \bigcup_{j=1}^k T_j$
is either atoroidal or a Seifert manifold,
and this family is unique up to isotopy.
}
\end{itemize}
Recall that an \emph{essential} torus is a torus
which is embedded in $M$, two-sided, incompressible, 
and not boundary-parallel.
The components of $M \smallsetminus \bigcup_{j=1}^k T_j$
are called the \emph{pieces}.
As a corollary of this theorem, we have
\begin{itemize}
\item[$\bullet$]
\emph{If $M$ and $T_1, \hdots, T_k$ are as above, 
there is a graph of groups
$\mathbf G  = (G,Y)$,
with one vertex for each piece 
and one geometric edge for each torus,
and with $G_e = \pi_1(\mathbf T^2) \approx \mathbf Z^2$
for all $e \in E(Y)$,
such that $\pi_1(M)$ is isomorphic 
to the fundamental group of $\mathbf G$.}
\par
\emph{Moreover, if $k > 0$, 
either $M$ is a torus bundle over $\mathbf S^1$, and $k=1$,
or, for each edge $e \in E(Y)$,
the images of $G_e$ by $\varphi_e$ and $\varphi_{\overline{e}}$
are proper subgroups of $G_{t(e)}$ and $G_{s(e)}$ respectively,
and $k \ge 2$.
}
\end{itemize}
The last statement follows from Theorem 10.2 in
\cite{Hemp--76}.

\vskip.2cm

(5.6)
An orientable $3$-manifold $M$ is \emph{geometric} 
if the interior of $M$ is homeomorphic 
to a Riemannian manifold $\Lambda \backslash X$,
where $X$ is in a standard list of eight 3-dimensional manifolds,
traditionally denoted by
\begin{equation*}
\mathbf S^3, \hskip.2cm
\mathbf E^3, \hskip.2cm
\mathbf H^3, \hskip.2cm
\mathbf S^2 \times \mathbf R,  \hskip.2cm
\mathbf H^2 \times \mathbf R,  \hskip.2cm
\widetilde{PLS_2(\mathbf R)},  \hskip.2cm
 Nil \hskip.3cm \text{and} \hskip.3cm Sol ,
\end{equation*}
and where $\Lambda$ is  a group of isometries of $X$
acting freely 
on $X$.
Equivalently (this is a  theorem),
a $3$-manifold is geometric if its interior
admits a complete
Riemannian metric
which is  locally homogeneous,
namely which is such that 
any pair of points in this interior has a pair
of isometric neighbourhoods.

If $X$ is one of these eight ``models''
which is neither $\mathbf H^3$ nor $Sol$
then $M$ is necessarily a Seifert manifold.
It follows that a geometric $3$-manifold is of one of three types:
\begin{itemize}
\item[$\bullet$] an hyperbolic manifold,
\item[$\bullet$] a Seifert manifold,
\item[$\bullet$] a Sol manifold.
\end{itemize}
There are many $3$-manifolds which \emph{are not} geometric:
for example, a connected sum of two closed manifolds
is geometric if and only if it is 
the connected sum of two copies of $\mathbf P^3$.
For all this, see  \cite{Scot--83}, 
in particular Page 403 for the last claim,
as well as \cite{Bona--02}.

\vskip.2cm 

(5.7)
An oriented $3$-manifold is said to satisfy 
\emph{Thurston's Geometrisation Conjecture}
if the  pieces of the decomposition (5.5)
are geometric.
\par

In three preprints made public in 2002 and 2003, 
Grisha Perelman, following the Hamilton program and
using the Ricci flow, has sketched a proof 
of  Thurston's Geometrisation Conjecture;
particular cases have been known before.
See \cite{MoTi} and references there. 



\section{C$^*$-simplicity of $3$-groups: 
\\
proofs of Propositions \ref{orientable} % 8
to \ref{nonorientable} % 10
}
% 6
\label{section6}

We denote by $\mathbf K^2 \widetilde{\times} \mathbf I$
the non-trivial $\mathbf I$-bundle over the Klein bottle. 

\begin{Lemma}
% 29
\label{1duchap6}
Let $N$ be a compact orientable connected  $3$-manifold
which is $\partial$-irreducible 
and with a connected component $T$ of the boundary
which is a $2$-torus.
Assume that the fundamental group $\pi_1(N)$
contains a normal subgroup $K \ne \{1\}$
which is peripheral, in the image of $\pi_1(T)$.
\par
Then one of the following conclusion holds:
\begin{itemize}
\item[(i)]
$K \approx \mathbf Z$ and $N$ is a Seifert manifold,
\item[(ii)]
$K \approx \mathbf Z^2$ and $N \approx \mathbf T^2 \times \mathbf I$,
\item[(iii)]
$K \approx \mathbf Z^2$ and $N \approx \mathbf K^2 \widetilde{\times} \mathbf I$.
\end{itemize}
\end{Lemma}

\noindent
\emph{Proof.}
Observe that $K$, as a subgroup of $\pi_1(T)$,
is isomorphic to one of $\mathbf Z$ or $\mathbf Z^2$.
Consider two cases, depending on $Q \Doteq \pi_1(N)/K$
is finite or not.
\par

\emph{Case 1:  $Q$ is finite.}
Assume first that $K \approx \mathbf Z^2$.
Theorem 10.6 in \cite{Hemp--76} implies
that $N$ is a $\mathbf I$-bundle over a surface $F$;
since $F$ is necessarily homeomorphic 
to one of $\mathbf T^2$ or $\mathbf K^2$,
one of (ii), (iii) holds.
\par
Assume next  that $K \approx \mathbf Z$;
we will arrive at a contradiction.
By a theorem of Epstein (9.8 in \cite{Hemp--76}), 
$\pi_1(N)$ is torsion-free.
Since $\pi_1(N)$ contains an infinite cyclic subgroup of finite index,
$\pi_1(N)$ is itself infinite cyclic.
But $\pi_1(N)$ contains $\pi_1(T)$ which is isomorphic to $\mathbf Z^2$,
and this is impossible.
\par

\emph{Case 2: $Q$ is infinite.}
By Consequence (3) in Theorem 11.1 of \cite{Hemp--76},
we have $K \approx \mathbf Z$.
By the theorem of Seifert fibration
(see for our case Theorem II.6.4 in \cite{JaSh--79}),
$N$ is a Seifert manifold.\hfill $\square$

\vskip.2cm

Let $M$ be a  orientable connected compact $3$-manifold
which is not simply connected;
we assume that $\mathcal P (M) = M$
(or equivalently, modulo the Poincar\'e Conjecture,
that $\widehat M = M$).
We want to prove Proposition~\ref{orientable};
by Proposition \ref{3manifoldtrichotomy},
we can assume that $M$ is moreover $\partial$-irreducible.
\par

If  $M$ is geometric,
then $M$ is either hyperbolic, or Seifert, or Sol,
and Proposition \ref{orientable} % 8
for these cases follows from Example~\ref{Ex1a4}. % 7.
Thus, we can assume furthermore that 
\begin{itemize}
\item[$\bullet$]
\emph{$M$ is not geometric and $M$ contains an essential $2$-torus $T$;} 
\end{itemize}
see (5.5).
Proposition \ref{orientable} % 8
follows now from the two next lemmas.

\vskip.2cm

\begin{Lemma}
% 30
\label{ex6.1}
In the previous situation, if $T$ is separating, 
then $\Gamma$ is a strongly Powers group.
\end{Lemma}

\noindent
\emph{Proof.}
Denote by $M_1, M_2$ 
the two components of the result of splitting $M$ along $T$.
Since $M$ is $\partial$-irreducible and $T$ is essential,
$M_1$ and $M_2$ are also $\partial$-irreducible.
Set $A = \pi_1(M_1)$, $B = \pi_1(M_2)$, 
and $C = \pi_1(T) \approx \mathbf Z^2$,
so that $\Gamma = A \ast_C B$
by the Seifert -- Van Kampen theorem;
the amalgam is non-trivial, see (5.5). 
By Proposition \ref{C*simpleamalg},  % 2
either $\Gamma$ is a strongly Powers group,
or $C_k \ne \{1\}$ for all $k \ge 1$.
From now on, we will assume that $C_2 \ne \{1\}$
(so that $C_1 \ne \{1\}$),
and we will obtain a contradiction.
\par

Set $C_A = \bigcap_{a \in A} a^{-1} C a$ 
and $C_B = \bigcap_{b \in B} b^{-1} Cb$,
so that $C_1 = C_A \cap C_B$
is a non-trivial subgroup of $C \approx \mathbf Z^2$.
Observe that $C_A$ is isomorphic to 
either $\mathbf Z$ or $\mathbf Z^2$, 
and also that $C_A$ is the largest subgroup of $C$
which is normal in $A$.
\par

Note that neither $M_1$ nor $M_2$ can be
homeomorphic to $\mathbf T^2 \times \mathbf I$,
otherwise $T$ would be boundary parallel in $M$,
and this is not the case by (5.5).
Hence each of $M_1, M_2$
is either as in (i) or as in (iii)
of Lemma \ref{1duchap6}, % 29
and we will end the proof by showing that each case is ruled out.
\par

Suppose first that 
$M_1 \approx M_2 \approx \mathbf K^2 \widetilde{\times} \mathbf I$.
Then $\partial M_1 \approx \partial M_2 \approx \mathbf T^2$,
and we have two $2$-coverings $N_j \longrightarrow M_j$
with $N_j \approx \mathbf T^2 \times \mathbf I$ ($j=1,2$).
Thus we have a $2$-covering $N \longrightarrow M$
where $N = N_1 \cup N_2$ is a $\mathbf T^2$-bundle over $\mathbf S^1$,
so that $N$ can be viewed as obtained from
$\mathbf T^2 \times \mathbf I$ by identifying 
the two connected components of the boundary
by a gluing homeomorphism  of $\mathbf T^2$.
A torus bundle over $\mathbf S^1$ is 
Sol if the gluing map is Anosov,
and Seifert if the gluing map is periodic or reducible
(Theorem 5.5 in \cite{Scot--83} 
or Exercice 3.8.10 in \cite{Thur--97}). 
If $N$ is Sol, then  $M$ is also Sol
(Theorem 5.3.i in \cite{Scot--83}).
If  $N$ is Seifert, then  $M$ is also Seifert
(Theorem II.6.3 in \cite{JaSh--79}).
Thus, in all cases, $M$ is geometric
(see Page 483 of \cite{Scot--83}),
but this has been ruled out.
Thus this case does not occur.
\par

Suppose now that $M_1$ and $M_2$ are Seifert,
with 
\begin{equation*}
A \,  \rhd \, C_A \, \approx \, \mathbf Z \, \approx \, C_B \, \lhd \, B .
\end{equation*}
Since any subgroup of $\mathbf Z$ is characteristic,
$C_1 = C_A \cap C_B$ is normal in both $A$ and $B$,
so that $C_1 \approx \mathbf Z$ is also normal in $\Gamma$.
By the theorem of Seifert fibrations 
(already used in the proof of Lemma \ref{1duchap6}), % 29
it follows that $M$ is Seifert, but this has been ruled out.
\par

We can finally suppose that $M_1$ is Seifert
with $Z \approx C_A \lhd A$,
and $M_2 = \mathbf K^2 \widetilde{\times} \mathbf I$
with $\partial M_2 \approx \mathbf T^2$
and $C = \pi_1(\partial M_2) \approx \mathbf Z^2$;
since $C_B$ is the maximal subgroup of $C$
which is normal in $B$, we have $C_B = C$.
It follows that $C_1 = C_A \cap C_B = C_A$.
Since $C_1$ is normal in $A$,
\begin{equation*}
\{1\} \, \ne \, C_2 \, = \, 
C_1 \cap \Big( \bigcap_{b \in B} b^{-1} C_1 b \Big) \, \lhd \,
C_A \, \approx \, \mathbf Z ,
\end{equation*}
$C_2$ is normal in both $A$ and $B$,
and therefore also in $\Gamma$.
(Though we will not use this, let us observe that
$C_k = C_2$ for al $k \ge 2$.)
It follows as above that $M$ is Seifert,
which has been ruled out.
\hfill $\square$


\begin{Lemma}
% 31
\label{ex6.2}
In the previous situation, if $T$ is non-separating, 
then $\Gamma$ is a strongly Powers group.
\end{Lemma}

\noindent
\emph{Proof.}
To prove this claim, denote by $M_1$
the result of splitting $M$ along $T$.
Since $M$ is $\partial$-irreducible and $T$ is essential,
$M_1$ is also $\partial$-irreducible.
Set $G = \pi_1(M_1)$;
the two boundary components of $M_1$ coming from the splitting
correspond to two isomorphic subgroups of $G$, 
say $H$ and $\theta(H)$ where $\theta$ 
is an isomorphism with domain $H$,
and $H \approx \pi_1(T) \approx \mathbf Z^2$.
If one had $H = G$ or $\theta (H) = G$,
the manifold $M$ would be a $\mathbf T^2$-bundle 
over $\mathbf S^1$ (Theorem 10.2 in \cite{Hemp--76}), 
and this cannot be (as above).
It follows that the HNN-extension
$\Gamma = \pi_1(M) = HNN(G,H,\theta)$
is non-degenerate.
By Proposition \ref{C*simpleHNN}, % 4
either $\Gamma$ is a strongly Powers group,
or $H_k \ne \{1\}$ for all $k \ge 1$.
From now on, we will assume that $H_2 \ne \{1\}$
(so that $H_1 \ne \{1\}$),
we will obtain a contradiction.
\par

Set $H_G = \bigcap_{g \in G} g^{-1} H g$,
so that $H_1 = H_G \cap \tau^{-1} H \tau \cap \tau H \tau^{-1}$
is a non-trivial subgroup of $H \approx \mathbf Z^2$.
Observe that $H_G$ is isomorphic to
either $\mathbf Z$ or $\mathbf Z^2$,
and also that $H_G$ is the maximal subgroup of $H$
which is normal in $G$.
\par

The boundary of $M_1$ has at least 
two connected components which are $2$-tori. Since 
$\partial(\mathbf K^2 \widetilde{\times} \mathbf I) \approx \mathbf T^2$
is connected, $M_1$ cannot fit in Case (iii) of Lemma \ref{1duchap6}.
If one had $H_G \approx \mathbf Z^2$ 
and $M_1 \approx \mathbf T^2 \times \mathbf I$, 
the manifold $M$ would be a $\mathbf T^2$-bundle over $\mathbf S^1$,
consequently would be geometric, and this is ruled out (as above).
\par

Therefore, because of Lemma \ref{1duchap6},
we can assume that $M_1$ is Seifert and that $H_G \approx \mathbf Z$.
Choose a generator $h$ of $H_1$, so that $H_1 = \langle h \rangle$.
Since $\langle h \rangle$ is characteristic in $H_G$,
the subgroup $\langle h \rangle$ is normal in $G$.
Since
\begin{equation*}
\{1\} \, \ne \, H_2 \, = \, 
\langle h \rangle \cap \tau^{-1} \langle h \rangle \tau \cap
\tau \langle h \rangle \tau^{-1} ,
\end{equation*}
there exists a pair of non-zero integers $p,q$ such that
$\tau^{-1} h^p \tau = h^q$.
By what we know about Baumslag-Solitar subgroups
of $3$-manifold groups (Theorem VI.2.1 in \cite{JaSh--79}),
this implies that $q = \pm p$.
Hence $\Gamma$ contains a normal subgroup
$\langle h^p \rangle \approx \mathbf Z$.
It follows as above that $M$ is a Seifert manifold,
in particular a geometric manifold, and this has been ruled out.
\par

This ends the proof.
\hfill $\square$

\vskip.2cm

\noindent
\textbf{Proof of Corollary \ref{knot}.} 
Let $K$ be a knot in $\mathbf S^3$.
Denote by $M$ the complement of an open tubular neighbourhood of $K$,
and let $\Gamma = \pi_1(M)$ the group of $K$.
Then $M$ is irreducible, by the Alexander-Sch\"onflies Theorem.
Assume moreover that $K$ is not trivial, 
so that $\partial M$ is an incompressible torus 
(Proposition 3.17  in \cite{BuZi--85}),
and $M$ a $\partial$-irreducible manifold.
\par

Suppose first that the JSJ decomposition of $M$ is trivial,
so that $M$ is either Seifert or atoroidal (see 5.5).
If $M$ is a Seifert manifold, then 
$\Gamma$ is not C$\, ^*$-simple
beause $K$ is a \emph{torus knot};
see \cite{Budn--06}, Proposition 4, first case, with $n=1$
(observe that, in the second case of the same proposition,
$n \ge 1$ should be replaced by $n \ge 2$).
If $M$ is atoroidal, Thurston's Hyperbolisation Theorem implies
that $M$ is an hyperbolic manifold of finite volume
(see e.g. \cite{Bona--02}, Section 6.1); 
in this case, $K$ is a \emph{hyperbolic knot} and
$\Gamma$ is a strongly Powers group,
see Example ~\ref{Ex1a4}. % 7
\par

Suppose now that the JSJ decomposition of $M$ is non-trivial
(i.e. involves at least one torus),
so that $K$ is a \emph{satellite knot}.
Geometric manifolds of this kind are classified;
they are special kinds of Sol manifolds
(see Theorem 2.11 in \cite{Bona--02})
and cannot be knot complements.
Hence $M$ is not geometric, 
and $\Gamma$ is a strongly Powers group by our 
Lemmas \ref{ex6.1} and \ref{ex6.2}. % 30 et % 31
\hfill $\square$

\hskip.2cm

\noindent
\textbf{Proof of Propositions  \ref{nonorientable}.}
% 10
If $\Gamma$ is of order two,
neither $\Gamma$ nor $\Gamma' = \{1\}$ is C$^*$-simple,
and the Proposition holds for a trivial reason.
We assume from now on that $\Gamma$ is not of order two,
and it follows that $\Gamma$ is infinite
(a result of D. Epstein,  Theorem 9.5 in \cite{Hemp--76}).

Any subgroup of finite index in a C$^*$-simple group 
is itself C$^*$-simple \cite{BeHa--00}.
In particular, if $\Gamma$ is C$^*$-simple, so is $\Gamma'$.

Assume finally that $\Gamma'$ is C$^*$-simple.
Then $\Gamma'$ is icc (see Appendix J in \cite{Harp--07}).
It follows from Lemma 9.1 in \cite{HaPr--07} that $\Gamma$ is icc,
and then from \cite{BeHa--00} that $\Gamma$ is C$^*$-simple.
\hfill $\square$



\section{A reminder on Powers groups and C$^*$-simple groups}
\label{section7}
% 7

A group $\Gamma$ is a \emph{Powers groups}
if it is not reduced to one element and if, 
for any finite subset $F$ in $\Gamma \smallsetminus \{1\}$ 
and for any integer  $N \ge 1$, 
there exists a partition $\Gamma = C \sqcup D$ and elements
$\gamma_1,\hdots,\gamma_N$ in $\Gamma$ such that   
\begin{itemize}
\item[]
   $fC \cap C = \emptyset$ for all $f \in F$,
\item[]
   $\gamma_jD \cap \gamma_kD = \emptyset$ 
          for all $j,k \in \{1,\hdots,N\}$, $j \neq k$.
\end{itemize} 
Let us recall here the following facts:
\begin{itemize}
\item[$\bullet$]
non-abelian free groups are Powers groups \cite{Powe--75};
\item[$\bullet$]
Powers group have non-abelian free subgroups \cite{BrPi};
\item[$\bullet$]
a Powers group is C$^*$-simple, 
and moreover its reduced C$^*$-algebra has a unique trace;
\item[$\bullet$]
Proposition \ref{Powersgen} %17
applies to many examples for showing them to be Powers groups;
\item[$\bullet$]
there are uncountably many countable groups $\Gamma$
with pairwise non-isomorphic reduced C$^*$-algebras
(Corollary~9 in \cite{AkLe--80}, building up on \cite{McDu--69});
\item[$\bullet$]
if $N$ is an amenable normal subgroup of a Powers group
(or more generally of a C$^*$-simple group),
then $N = \{1\}$;
\item[$\bullet$]
there are  C$^*$-simple groups which are not Powers groups
(examples: $PSL_n(\mathbf Z)$ for large $n$, and
direct products of non-abelian free groups);
\end{itemize}
\par\noindent
see  \cite{Harp--07}.
We do not know any example of a Powers group 
which is not a strongly Powers group, as defined in the introduction;
indeed, the following question is open for us:
\begin{itemize}
\item[$\bullet$]
\emph{does there exist a pair $(\Gamma,N)$
of a group and a non-trivial normal subgroup
with $\Gamma$ C$\, ^*$-simple
and $N$ not C$\, ^*$-simple?}
\end{itemize}
We are grateful to Bachir Bekka who has observed to us that,
as a consequence of [Pozn], if such a pair $(\Gamma,N)$ exists,
then $N$ cannot be linear (\emph{a fortiori} $\Gamma$ cannot be linear).
Indeed, suppose that a C*-simple  group $\Gamma$
would contain a linear non-C*-simple non-trivial normal subgroup $N$; 
then, by Poznansky's results, the amenable radical $R$ of $N$ would be non-trivial;
but amenable radicals are characteristic subgroups, 
so that $N$ would be normal in $\Gamma$,
and this is impossible since $\Gamma$ is C*-simple. 
The question is closely related to that about the existence
of a group $\Gamma$ which would \emph{not}  be C$^*$-simple
and which would not contain any non-trivial amenable normal subgroup
\cite{BeHa--00}.

\par

Finally, let us state the following strenghtening of
Proposition~11 of \cite{Harp--07}.
Given a group $\Gamma$ of homeomorphisms of a topological space $\Omega$,
recall that the subset $L_{\Gamma}$ of $\Omega$ has been defined
in the proof of Proposition \ref{Powersgen}. % 17


\begin{Prop}
% 32
\label{11deHarp--07}
Let $\Gamma$ be a group which acts by homeomorphisms on
a Hausdorff topological space $\Omega$. 
Assume the action of $\Gamma$ is strongly hyperbolic on $\Omega$
and strongly faithful on $L_{\Gamma}$. Then
\begin{itemize}
\item[(i)]
any non-empty $\Gamma$-invariant  closed subset of $\Omega$
contains $\overline{L_{\Gamma}}$.
\end{itemize}
Let moreover $N$ be a normal subgroup of $\Gamma$
which contains an hyperbolic homeomorphism. Then
\begin{itemize}
\item[(ii)]
$\overline{L_N} = \overline{L_{\Gamma}}$;
\item[(iii)]
$N$ is a Powers group.
\end{itemize}
\end{Prop}

\noindent 
\emph{Proof.}
Observe that $L_{\Gamma}$ is non-empty
because $\Gamma$ contains hyperbolic elements,
and indeed infinite because the action of $\Gamma$
is strongly hyperbolic.
Claim (i) follows from the argument used for
Proposition \ref{minimal}, % 16
and Claim (ii) is then straightforward.
Thus the action of $N$ on $\Omega$
satisfies the hypothesis of Proposition 11 in [Harp--07], 
so that Claim~(iii) follows.
\hfill $\square$

\hskip.2cm


Let $M$ be a connected $3$-manifold which is hyperbolic and not elementary,
as in Example \ref{Ex1a4}. % 7
Let $\Gamma = \pi_1(M)$ be acting in the canonical way
on $\Omega = \mathbf S^2$, 
the boundary of the classical hyperbolic space $\mathbf H^3$ of dimension $3$.
Since $\Gamma$ is discrete on $\mathbf H^3$ and not elementary,
$\Gamma$ is strongly hyperbolic on $\mathbf S^2$ 
and strongly faithful on $L_{\Gamma}$.
It is then standard that 

\begin{Cor}
% 33
With the notation above, 
any normal subgroup $N \ne \{1\}$ of $\Gamma = \pi_1(M)$
contains transverse pairs of hyperbolic transformations.
\end{Cor}

\noindent \emph{Proof.}
On the one hand, $N$ cannot contain non-identity elliptic transformations,
because $\Gamma$ is torsion-free.
On the other hand,
if $N$ contains parabolic transformations,
then the closure of
\begin{equation*} 
\{ \eta \in \mathbf S^2
\hskip.1cm \vert \hskip.1cm
\gamma \eta = \eta 
\hskip.2cm \text{for some parabolic} \hskip.2cm
\gamma \in \Gamma \} .
\end{equation*}
coincides with $\overline{L_{\Gamma}}$ 
by (i) of the previous proposition,
so that $N$ contains pairs $(p_1, p_2)$ of parabolic elements
with distinct fixed points,
and $p_1^k p_2^k$ is hyperbolic for $k$ large enough.
\par

Thus $L_N$ is non-empty. Since $L_N$ is $\Gamma$-invariant,
$\overline{L_N} = \overline{L_\Gamma}$,
and the conclusion follows.
\hfill $\square$

\vskip.2cm

Similarly, in a  Gromov-hyperbolic group,
any infinite subnormal subgroup contains hyperbolic transformations.

  

\begin{thebibliography}{BMMN--98}

\bibitem[AkLe--80]{AkLe--80}
Charles A. \textsc{Akemann} and Tan-Xu \textsc{Lee},
\emph{Some simple C$\,{}^*$-algebras associated with free groups},
Indiana Univ. Math. J. \textbf{29} (1980), 505--511.

\bibitem[Bass--76]{Bass--76}
Hyman \textsc{Bass},
\emph{Some remarks on group actions on trees},
Communications in Algebra \textbf{4}:12 (1976), 1091--1126.

\bibitem[Bass--93]{Bass--93}
Hyman \textsc{Bass},
\emph{Covering theory for graphs of groups},
J. Pure and Applied Algebra \textbf{89} (1993), 3--47.

\bibitem[Bedo--84]{Bedo--84}
Erik \textsc{B\'edos},
\emph{Operator algebras associated with 
free products of groups with amalgamation},
Math. Annalen \textbf{266} (1984), 279--286.

\bibitem[BeHa--86]{BeHa--86}
Erik \textsc{B\'edos} and Pierre \textsc{de la Harpe},
\emph{Moyennabilit\'e int\'erieure des groupes~:
d\'efinitions et exemples},
L'Enseignement Math. \textbf{32} (1986), 139--157.

\bibitem[BeHa--00]{BeHa--00}
Bachir \textsc{Bekka} and Pierre \textsc{de la Harpe},
\emph{Groups with simple reduced C$\,{}^*$-algebras},
Expositiones Math. \textbf{18} (2000), 215--230.

\bibitem[BMMN--98]{BMMN--98}
Meenaxi \textsc{Bhattacharjee}, Dugald \textsc{Macpherson}, 
R\"ognvaldur G. \textsc{M\"oller}, and Peter M. \textsc{Neumann},
\emph{Notes on infinite permutation groups},
Lecture Notes in Math. \textbf{1698}, Springer, 1998.

\bibitem[Bona--02]{Bona--02}
Francis \textsc{Bonahon},
\emph{Geometric structures on $3$-manifolds},
in ``Handbook of geometric topology'',
R.J. Daverman and R.B. Sher Editors, Elsevier 2002,
pages 93--164.

\bibitem[BoNi--88]{BoNi--88}
Florin \textsc{Boca} and Viorel \textsc{Nitica},
\emph{Combinatorial properties of groups and
simple C$\,{}^*$-algebras with a unique trace},
J. Operator Theory \textbf{20} (1988), 183--196.

\bibitem[Bou-TG9]{Bou-TG9}
Nicolas \textsc{Bourbaki},
\emph{Topologie g\'en\'erale, chapitre~9,
utilisation des nombres r\'eels en topologie g\'en\'erale},
deuxi\`eme \'edition, Hermann, 1958.

\bibitem[BrHa--99]{BrHa--99}
Martin R. \textsc{Bridson} and Andr\'e \textsc{Haefliger},
\emph{Metric spaces of non-positive curvature},
Grundlehren der mathematischen Wissenschaften \textbf{319},
Springer, 1999.



\bibitem[BrPi]{BrPi} Matthew \textsc{Brin} 
and Gabriel \textsc{Picioroaga},
\emph{Unpublished note},
reproduced just after Question 15 in \cite{Harp--07}.

\bibitem[Budn--06]{Budn--06}
Ryan \textsc{Budney},
\emph{JSJ-decompositions of knot and link complements in $\mathbf S^3$},
L'Enseignement Math\'ematique \textbf{52} (2006), 319--359.

\bibitem[BuZi--85]{BuZi--85}
G.~ \textsc{Burde} and H.~ \textsc{Zieschang},
\emph{Knots},
de Gruyter, 1985.
 
\bibitem[Corn--09]{Corn--09}
Yves \textsc{de Cornulier},
\emph{Infinite conjugacy classes in groups acting on trees},
Groups Geom. Dyn. \textbf 3 (2009), 267--277.

\bibitem[Davi--96]{Davi--96}
Kenneth R. \textsc{Davidson},
\emph{C$\,{}^*$-algebras by example},
Amer. Math. Soc., 1996.

\bibitem[HaPr--07]{HaPr--07} 
Pierre \textsc{de la Harpe} and Jean-Philippe 
\textsc{Pr\'eaux}, \emph{Groupes fondamentaux des
vari\'et\'es de dimension 3 et alg\`ebres d'op\'erateurs}, 
Annales de la Facult\'e des Sciences de Toulouse
{\bf 16} 
% (3), 
(2007),  561-589.

\bibitem[Harp--85]{Harp--85}
Pierre \textsc{de la Harpe},
\emph{Reduced C$\,{}^*$-algebras of discrete groups 
which are simple with a unique trace},
in \emph{Operator algebras and their connections 
with topology and ergodic theory
(Busteni, 1983)}, Lecture Notes in Math. \textbf{1132} (1985), 230--253.


\bibitem[Harp--07]{Harp--07}
Pierre \textsc{de la Harpe},
\emph{On simplicity of reduced C$\,{}^*$-algebras of groups},
Bull. London Math. Soc. \textbf{39}:1 (2007), 1--26.

\bibitem[Hemp--76]{Hemp--76} 
John \textsc{Hempel}, 
\emph{$3$-manifolds}, 
Princeton University Press, 1976.

\bibitem[Ivan]{Ivan}
Nikolay A. \textsc{Ivanov},
\emph{On the structure of some reduced amalgamated
free product C$\,{}^*$-algebras},
arXiv.0705.3919v3 (4 Sep 2007).

\bibitem[JaSh--79]{JaSh--79} 
William H. \textsc{Jaco} and Peter B. \textsc{Shalen}, 
\emph{Seifert fibered spaces in $3$-manifolds}, 
Memoirs of the Amer. Math. Soc., vol 21, {\bf 220} (1), 1979.

\bibitem[LySc--77]{LySc--77}
Roger C. \textsc{Lyndon} and Paul E. \textsc{Schupp},
\emph{Combinatorial group theory},
Springer, 1977.

\bibitem[McDu--69]{McDu--69}
Dusa \textsc{McDuff},
\emph{Uncountably many $II_1$ factors},
Annals of Math. \textbf{90} (1969), 372--377.

\bibitem[Mold--91]{Mold--91}
D. \textsc{Moldavanskii},
\emph{On the isomorphisms of Baumslag-Solitar groups},
Ukrainian Math. J. \textbf{43}:12 (1991), 1569--1571.

\bibitem[MoTi]{MoTi}
John \textsc{Morgan} and Gang \textsc{Tian},
\emph{Completion of the proof of the geometrization conjecture},
arXiv:0809.4040v1    (23 Sep 2008).

\bibitem[MoSh--04]{MoSh--04}
Nicolas \textsc{Monod} and Yehuda \textsc{Shalom},
\emph{Cocycle superrigidity and bounded cohomology
for negatively curves spaces},
J. Differential Geometry \textbf{67} (2004), 395--455.

\bibitem[NeBH--54]{NeBH--54}
Bernhard Hermann \textsc{Neumann}, 
\emph{Groups covered by finitely many cosets},
Publ. Math. Debrecen \textbf{3} (1954), 227--242.

\bibitem[NePM--76]{NePM--76}
Peter Michael \textsc{Neumann},
\emph{The structure of finitary permutation groups},
Arch. Math. \textbf{27} (1976), 3--17.

\bibitem[PaSa--79]{PaSa--79}
William L. \textsc{Paschke} and Norberto \textsc{Salinas},
\emph{C$\,{}^*$-algebras associated with the free products of groups},
Pacific J. Math. \textbf{82} (1979), 211--221.

\bibitem[PaVa--91]{PaVa--91}
Isabelle \textsc{Pays} and Alain \textsc{Valette},
\emph{Sous-groupes libres dans les groupes d'automorphismes d'arbres},
L'Enseignement Math\'ematique \textbf{37} (1991), 151--174.

\bibitem[Powe--75]{Powe--75}
Robert T. \textsc{Powers},
\emph{Simplicity of the C$\,{}^*$-algebra 
associated with the free group on two generators},
Duke Math. J. \textbf{42} (1975), 151--156.

\bibitem[Pozn]{Pozn}
Tal \textsc{Poznansky},
\emph{Characterization of linear groups whose reduced C$^*$-algebras are simple},
arXiv:0812.2486vt   (24 May 2009).

\bibitem[Rham--69]{Rham--69}
Georges \textsc{de Rham},
\emph{Lectures on introduction to algebraic topology},
Tata Institute of Fundamental research, Bombay (1969).

\bibitem[Scot--83]{Scot--83} 
Peter \textsc{Scott}, 
\emph{The geometries of $3$-manifolds}, Bull. London Math. Soc. {\bf 15}
(1983),  401-487.


\bibitem[ScWa--79]{ScWa--79}
Peter \textsc{Scott} and Terry \textsc{Wall}
\emph{Topological methods in group theory},
in ``Homological group theory, Durham 1977'', C.T.C. Wall Editor,
Cambridge Univ. Press (1979), 137--203.

\bibitem[Sela--97]{Sela--97}
Zlil \textsc{Sela},
\emph{Acylindrical accessibility for groups},
Inventiones Math. \textbf{129} (1997), 527--565.


\bibitem[Serr--77]{Serr--77}
Jean-Pierre \textsc{Serre},
\emph{Arbres, amalgames, $SL_2$},
Ast\'erisque \textbf{46},
Soc. Math. France, 1977.

\bibitem[Souc--01]{Souc--01}
Estelle \textsc{Souche},
\emph{Quasi-isom\'etries et quasi-plans 
dans l'\'etude des groupes discrets},
Th\`ese, Universit\'e de Provence,
17 d\'ecembre 2001.

\bibitem[Stal--06]{Stal--06}
Yves \textsc{Stalder},
\emph{Moyennabilit\'e int\'erieure et extensions HNN},
Ann. Inst. Fourier \textbf{56}:2 (2006), 309--323.

\bibitem[Tao--62]{Tao--62}
Yoko \textsc{Tao},
\emph{On fixed point free involutions of $S^1 \times S^2$},
Osaka Math. J. \textbf{14} (1962), 145--152.

\bibitem[Thur--97]{Thur--97} 
William P. \textsc{Thurston}, 
\emph{Three-Dimensional Geometry and Topology, vol.1},
Princeton University Press, Ed. Silvio Levy, 1997.

\bibitem[Tits--70]{Tits--70}
Jacques \textsc{Tits},
\emph{Sur le groupe des automorphismes d'un arbre},
in ``M\'emoires d\'edi\'es \`a Georges de Rham'', 
publi\'es par Andr\'e Haefliger et Raghavan Narasimhan
(Springer 1970), pages 188--211.

\end{thebibliography}


\end{document}






0909.3528.extracted_bib
\begin{thebibliography}{BMMN--98}

\bibitem[AkLe--80]{AkLe--80}
Charles A. \textsc{Akemann} and Tan-Xu \textsc{Lee},
\emph{Some simple C$\,{}^*$-algebras associated with free groups},
Indiana Univ. Math. J. \textbf{29} (1980), 505--511.

\bibitem[Bass--76]{Bass--76}
Hyman \textsc{Bass},
\emph{Some remarks on group actions on trees},
Communications in Algebra \textbf{4}:12 (1976), 1091--1126.

\bibitem[Bass--93]{Bass--93}
Hyman \textsc{Bass},
\emph{Covering theory for graphs of groups},
J. Pure and Applied Algebra \textbf{89} (1993), 3--47.

\bibitem[Bedo--84]{Bedo--84}
Erik \textsc{B\'edos},
\emph{Operator algebras associated with 
free products of groups with amalgamation},
Math. Annalen \textbf{266} (1984), 279--286.

\bibitem[BeHa--86]{BeHa--86}
Erik \textsc{B\'edos} and Pierre \textsc{de la Harpe},
\emph{Moyennabilit\'e int\'erieure des groupes~:
d\'efinitions et exemples},
L'Enseignement Math. \textbf{32} (1986), 139--157.

\bibitem[BeHa--00]{BeHa--00}
Bachir \textsc{Bekka} and Pierre \textsc{de la Harpe},
\emph{Groups with simple reduced C$\,{}^*$-algebras},
Expositiones Math. \textbf{18} (2000), 215--230.

\bibitem[BMMN--98]{BMMN--98}
Meenaxi \textsc{Bhattacharjee}, Dugald \textsc{Macpherson}, 
R\"ognvaldur G. \textsc{M\"oller}, and Peter M. \textsc{Neumann},
\emph{Notes on infinite permutation groups},
Lecture Notes in Math. \textbf{1698}, Springer, 1998.

\bibitem[Bona--02]{Bona--02}
Francis \textsc{Bonahon},
\emph{Geometric structures on $3$-manifolds},
in ``Handbook of geometric topology'',
R.J. Daverman and R.B. Sher Editors, Elsevier 2002,
pages 93--164.

\bibitem[BoNi--88]{BoNi--88}
Florin \textsc{Boca} and Viorel \textsc{Nitica},
\emph{Combinatorial properties of groups and
simple C$\,{}^*$-algebras with a unique trace},
J. Operator Theory \textbf{20} (1988), 183--196.

\bibitem[Bou-TG9]{Bou-TG9}
Nicolas \textsc{Bourbaki},
\emph{Topologie g\'en\'erale, chapitre~9,
utilisation des nombres r\'eels en topologie g\'en\'erale},
deuxi\`eme \'edition, Hermann, 1958.

\bibitem[BrHa--99]{BrHa--99}
Martin R. \textsc{Bridson} and Andr\'e \textsc{Haefliger},
\emph{Metric spaces of non-positive curvature},
Grundlehren der mathematischen Wissenschaften \textbf{319},
Springer, 1999.



\bibitem[BrPi]{BrPi} Matthew \textsc{Brin} 
and Gabriel \textsc{Picioroaga},
\emph{Unpublished note},
reproduced just after Question 15 in \cite{Harp--07}.

\bibitem[Budn--06]{Budn--06}
Ryan \textsc{Budney},
\emph{JSJ-decompositions of knot and link complements in $\mathbf S^3$},
L'Enseignement Math\'ematique \textbf{52} (2006), 319--359.

\bibitem[BuZi--85]{BuZi--85}
G.~ \textsc{Burde} and H.~ \textsc{Zieschang},
\emph{Knots},
de Gruyter, 1985.
 
\bibitem[Corn--09]{Corn--09}
Yves \textsc{de Cornulier},
\emph{Infinite conjugacy classes in groups acting on trees},
Groups Geom. Dyn. \textbf 3 (2009), 267--277.

\bibitem[Davi--96]{Davi--96}
Kenneth R. \textsc{Davidson},
\emph{C$\,{}^*$-algebras by example},
Amer. Math. Soc., 1996.

\bibitem[HaPr--07]{HaPr--07} 
Pierre \textsc{de la Harpe} and Jean-Philippe 
\textsc{Pr\'eaux}, \emph{Groupes fondamentaux des
vari\'et\'es de dimension 3 et alg\`ebres d'op\'erateurs}, 
Annales de la Facult\'e des Sciences de Toulouse
{\bf 16} 
% (3), 
(2007),  561-589.

\bibitem[Harp--85]{Harp--85}
Pierre \textsc{de la Harpe},
\emph{Reduced C$\,{}^*$-algebras of discrete groups 
which are simple with a unique trace},
in \emph{Operator algebras and their connections 
with topology and ergodic theory
(Busteni, 1983)}, Lecture Notes in Math. \textbf{1132} (1985), 230--253.


\bibitem[Harp--07]{Harp--07}
Pierre \textsc{de la Harpe},
\emph{On simplicity of reduced C$\,{}^*$-algebras of groups},
Bull. London Math. Soc. \textbf{39}:1 (2007), 1--26.

\bibitem[Hemp--76]{Hemp--76} 
John \textsc{Hempel}, 
\emph{$3$-manifolds}, 
Princeton University Press, 1976.

\bibitem[Ivan]{Ivan}
Nikolay A. \textsc{Ivanov},
\emph{On the structure of some reduced amalgamated
free product C$\,{}^*$-algebras},
arXiv.0705.3919v3 (4 Sep 2007).

\bibitem[JaSh--79]{JaSh--79} 
William H. \textsc{Jaco} and Peter B. \textsc{Shalen}, 
\emph{Seifert fibered spaces in $3$-manifolds}, 
Memoirs of the Amer. Math. Soc., vol 21, {\bf 220} (1), 1979.

\bibitem[LySc--77]{LySc--77}
Roger C. \textsc{Lyndon} and Paul E. \textsc{Schupp},
\emph{Combinatorial group theory},
Springer, 1977.

\bibitem[McDu--69]{McDu--69}
Dusa \textsc{McDuff},
\emph{Uncountably many $II_1$ factors},
Annals of Math. \textbf{90} (1969), 372--377.

\bibitem[Mold--91]{Mold--91}
D. \textsc{Moldavanskii},
\emph{On the isomorphisms of Baumslag-Solitar groups},
Ukrainian Math. J. \textbf{43}:12 (1991), 1569--1571.

\bibitem[MoTi]{MoTi}
John \textsc{Morgan} and Gang \textsc{Tian},
\emph{Completion of the proof of the geometrization conjecture},
arXiv:0809.4040v1    (23 Sep 2008).

\bibitem[MoSh--04]{MoSh--04}
Nicolas \textsc{Monod} and Yehuda \textsc{Shalom},
\emph{Cocycle superrigidity and bounded cohomology
for negatively curves spaces},
J. Differential Geometry \textbf{67} (2004), 395--455.

\bibitem[NeBH--54]{NeBH--54}
Bernhard Hermann \textsc{Neumann}, 
\emph{Groups covered by finitely many cosets},
Publ. Math. Debrecen \textbf{3} (1954), 227--242.

\bibitem[NePM--76]{NePM--76}
Peter Michael \textsc{Neumann},
\emph{The structure of finitary permutation groups},
Arch. Math. \textbf{27} (1976), 3--17.

\bibitem[PaSa--79]{PaSa--79}
William L. \textsc{Paschke} and Norberto \textsc{Salinas},
\emph{C$\,{}^*$-algebras associated with the free products of groups},
Pacific J. Math. \textbf{82} (1979), 211--221.

\bibitem[PaVa--91]{PaVa--91}
Isabelle \textsc{Pays} and Alain \textsc{Valette},
\emph{Sous-groupes libres dans les groupes d'automorphismes d'arbres},
L'Enseignement Math\'ematique \textbf{37} (1991), 151--174.

\bibitem[Powe--75]{Powe--75}
Robert T. \textsc{Powers},
\emph{Simplicity of the C$\,{}^*$-algebra 
associated with the free group on two generators},
Duke Math. J. \textbf{42} (1975), 151--156.

\bibitem[Pozn]{Pozn}
Tal \textsc{Poznansky},
\emph{Characterization of linear groups whose reduced C$^*$-algebras are simple},
arXiv:0812.2486vt   (24 May 2009).

\bibitem[Rham--69]{Rham--69}
Georges \textsc{de Rham},
\emph{Lectures on introduction to algebraic topology},
Tata Institute of Fundamental research, Bombay (1969).

\bibitem[Scot--83]{Scot--83} 
Peter \textsc{Scott}, 
\emph{The geometries of $3$-manifolds}, Bull. London Math. Soc. {\bf 15}
(1983),  401-487.


\bibitem[ScWa--79]{ScWa--79}
Peter \textsc{Scott} and Terry \textsc{Wall}
\emph{Topological methods in group theory},
in ``Homological group theory, Durham 1977'', C.T.C. Wall Editor,
Cambridge Univ. Press (1979), 137--203.

\bibitem[Sela--97]{Sela--97}
Zlil \textsc{Sela},
\emph{Acylindrical accessibility for groups},
Inventiones Math. \textbf{129} (1997), 527--565.


\bibitem[Serr--77]{Serr--77}
Jean-Pierre \textsc{Serre},
\emph{Arbres, amalgames, $SL_2$},
Ast\'erisque \textbf{46},
Soc. Math. France, 1977.

\bibitem[Souc--01]{Souc--01}
Estelle \textsc{Souche},
\emph{Quasi-isom\'etries et quasi-plans 
dans l'\'etude des groupes discrets},
Th\`ese, Universit\'e de Provence,
17 d\'ecembre 2001.

\bibitem[Stal--06]{Stal--06}
Yves \textsc{Stalder},
\emph{Moyennabilit\'e int\'erieure et extensions HNN},
Ann. Inst. Fourier \textbf{56}:2 (2006), 309--323.

\bibitem[Tao--62]{Tao--62}
Yoko \textsc{Tao},
\emph{On fixed point free involutions of $S^1 \times S^2$},
Osaka Math. J. \textbf{14} (1962), 145--152.

\bibitem[Thur--97]{Thur--97} 
William P. \textsc{Thurston}, 
\emph{Three-Dimensional Geometry and Topology, vol.1},
Princeton University Press, Ed. Silvio Levy, 1997.

\bibitem[Tits--70]{Tits--70}
Jacques \textsc{Tits},
\emph{Sur le groupe des automorphismes d'un arbre},
in ``M\'emoires d\'edi\'es \`a Georges de Rham'', 
publi\'es par Andr\'e Haefliger et Raghavan Narasimhan
(Springer 1970), pages 188--211.

\end{thebibliography}