On Triples of Ideal Chambers in $\typeA_2$-Buildings
Anne Parreau
We investigate the geometry in a real Euclidean building $\Es$ of type $\typeA_2$ of some simple configurations in the associated projective plane at infinity $\PP$, seen as ideal configurations in $\Es$, and relate it with the projective invariants (from the cross ratio on $\PP$). In particular we establish a geometric classification of generic triples of ideal chambers of $\Es$ and relate it with the triple ratio of triples of flags.
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\input{triples_01_mise_en_forme.def}
\input{triples_02_notations.def}


\begin{document}

\title{On Triples of Ideal Chambers in  $\typeA_2$-Buildings}
%

\author{Anne Parreau}
%

%

\address{
Université Grenoble I et CNRS, Institut Fourier\\ 
BP 74, 38402 St-Martin-d'Hères cedex, France.}

\email{Anne.Parreau@ujf-grenoble.fr}

%

%

\begin{abstract}
We investigate the geometry 
in a  real Euclidean building $\Es$ of type $\typeA_2$ 
of some simple configurations in the associated projective plane
at infinity $\PP$, seen as ideal configurations in $\Es$, and relate
it with the projective invariants (from the cross ratio on $\PP$).  In
particular we establish a geometric classification of generic triples
of ideal chambers of $\Es$ and relate it with the triple ratio of
triples of flags.
\end{abstract}


\maketitle





\input{triples_0_intro.tex}

\input{triples_1_prelim.tex}

\input{triples_2_basic_ideal_configs.tex}

\input{triples_3_triples_of_ideal_chambers.tex}

\input{triples_4_bibli.tex}






\end{document}

triples_0_intro.tex
%




\section*{Introduction}
%
The triples of objects in the boundaries of 
spaces $\Es$  with geometric structures 
%
are basic tools, 
for example in the study of surface group representations.
%
For instance, in the cas where $\Es=\HH^2$,
they may be used to define
the notion of Euler class  \cite{Gol80}, and
Penner-Thurston shear coordinates on the Teichmüller space.
%
In the case where  $\Es=\HH^2_\CC$, 
%
the  ideal triples
%
are classified by   Cartan's angular invariant 
(see for example \cite[\S  7.1]{GolBook}), 
%
and they may be used to
define Toledo's invariant and maximal representations,  
(see \cite{Tol89}. 
%
See for instance \cite{BIW10} for generalization to higher rank
Hermitian symmetric spaces $\Es$ and the link with triples in their
Shilov boundary.
%
 In hyperbolic geometry, the interplay between the geometry of
 the hyperbolic space $\Es$ 
 and the projective geometry of the associated
 projective line at infinity $\bordinf\Es$
 is fundamental, and invariants of ideal configurations
 are often defined using cross ratios.
%
For higher rank  symmetric spaces $\Es$ of type $\typeA_{\N-1}$ 
(e.g. corresponding to the group $\PGL_\N(\RR)$), 
ideal configurations in $\Es$ may be seen as configurations in the
projective space $\PP=\PP(\RR^\N)$.
In particular, ideal chambers of $\Es$ 
correspond to complete flags in $\PP$,
%
and generic pairs of flags (or generic $\N$-tuples of points)
in $\PP$ correspond to maximal flats in $\Es$.
%
This is still true in the non-Archimedean setting, 
i.e. for $\Es$ a Euclidean building of type $\typeA_{\N-1}$
 (replacing $\RR$ by a ultrametric valuated field $\KK)$.
%
Configurations in projective spaces have been widely studied and used.
%
In particular, triples of flags in $\PP(\RR^3)$ and 
their classical invariant, the triple ratio, 
are the basic building block to define generalized shearing coordinates
for higher Teichmüller space \cite{FoGoIHES} 
(representations of surface groups in $\G=\SL_3(\RR)$). 
%

%
In this article, we investigate the geometry 
in an Euclidean building $\Es$ of type $\typeA_2$ 
of some simple ideal configurations, 
mainly the  generic triples of ideal  chambers,
and the relationship  with their projective geometry 
in the projective plane $\PP$.
%
Our first motivation is to use it to study actions 
of punctured surface groups 
on $\typeA_2$-Euclidean buildings $\Es$, 
using ideal triangulations 
and a geometric interpretation in $\Es$ 
of Fock-Goncharov  parameters 
(see \cite{ParDegFG}).
%
The main result is a classification 
of ideal triples of chambers 
by the geometry of the naturally associated flats in $\Es$,
%
in relation with  their triple ratio as triples of flags in $\PP$.
%
In the case where $\Es$ is a real tree 
(e.g. a real building of type $\typeA_1$), 
any generic ideal triple
bounds a {\em tripod} in $\Es$, 
that is a convex subset consisting 
of union of three rays 
from a point $\xE\in\Es$ (the {\em center} of the tripod).
%
This is no longer the case in general
in higher rank buildings like $\typeA_2$ buildings, 
and many types of configurations are possible.  
%
A special case was studied by A. Balser, 
who established a caracterisation 
%
of triples of points in $\bordinf\Es$ 
bounding a tripod in $\Es$ \cite{Balser08} 
(and used it to study convex rank $1$ subsets in $\typeA_2$-buildings). 
%
We give here a complete and precise description.


%

%

 




We now get in more details.
%
Let $\Es$ be 
a real Euclidean building of (vectorial) type $\typeA_2$, 
i.e. with model flat the Euclidean plane
$\Aa = 
\{ \vA = (\vA_1,\vA_2,\vA_3) \in \RR^3 
         /\ \sum_\indN\vA_\indN=0
\}$, endowed with the finite reflection group $\W=\Sym_3$
(acting by permutation of the coordinates).
%
Note that $\Es$ is not necessarily discrete (simplicial) 
nor locally compact, and possibly exotic.
%

%
The boundary at infinity of $\Es$ may be identified with 
the incidence graph of an associated projective plane 
$\PP=\PP_\infty(\Es)$, 
equipped with  
a $\RR$-valued (additive) cross ratio (or projective valuation) 
$\geombir$ 
defined on quadruples of collinear points in $\PP$
\cite{Tits86}.
%
In the algebraic case, 
i.e. when 
$\Es$ is the Bruhat-Tits building $\Es(\KK^3)$ 
%
associated with $\PGL(\KK^3)$ 
%
for some  ultrametric field $\KK$, 
then $\PP$ is the classical projective plane $\PP(\KK^3)$ 
and $\geombir$ is the logarithm of the absolute value 
$\geombir=\log\abs{\Bir}$ 
of the usual $\KK$-valued cross ratio $\Bir$. 
We will then call $\geombir$ the {\em geometric} cross ratio 
and $\Bir$ the {\em algebraic} cross ratio to distinguish them.
%
Conventions on cross ratios are taken such that
$\Bir(\infty,-1,0\Z)=\Z$ (following \cite{FoGoIHES}).
%

%
We now turn to ideal triples of chambers.
%
Let  $\Tau=(\F_1,\F_2,\F_3)$ 
be a generic triple of chambers at infinity of $\Es$. 
%
We denote by $\F_\ii=(\p_\ii,\D_\ii)$ the corresponding flag of $\PP$,
with $\p_\ii$ the point and $\D_\ii$ the line.
%
The set $\{1,2,3\}$ of indices 
will be canonically identified  with $\ZZ/3\ZZ$.

%
In the algebraic case, $\PP=\PP(\KK^3)$, and 
generic triples of flags $(\F_1,\F_2,\F_3)$
are classified 
by one $\KK$-valued invariant,  
the ({\em algebraic}) triple ratio 
(see for example \cite[\S 9.4]{FoGoIHES}), that may be defined by:
$$  \Tri(\F_1,\F_2,\F_3) =   \Bir(\D_1, \p_1\p_2,
\p_1\p_{23},\p_1\p_3)$$
where $\p_\iijj=\D_\ii\cap \D_\jj$. 
%
We recall that it is invariant under cyclic permutations of 
$\Tau=(\F_1,\F_2,\F_3)$,
and reversing the order we get 
%
$\Tri(\ov{\Tau})=\Tri(\Tau)^{-1}$
where $\ov{\Tau}=(\F_3,\F_2,\F_1)$.



%
In the general case, we introduce 
an invariant for generic triples of flags in $\PP$,
generalizing the usual triple ratio
the {\em (geometric) triple ratio}, 
which still make sense then the building $\Es$ is exotic
(non algebraic), 
whereas the usual triple ratio is not defined anymore.
%
We define it as the  triple of following cross ratios in $\PP$,
obtained from the four lines $\D_1, \p_1\p_2, \p_1\p_{23}, \p_1\p_3$
by cyclic permutation of the three last one. 
%
\[
\begin{array}{rl}
  \geomtrio(\F_1,\F_2,\F_3) 
&=   \geombir(\D_1, \p_1\p_2, \p_1\p_{23}, \p_1\p_3)\\
%
  \geomtrip(\F_1,\F_2,\F_3)
&=  \geombir(\D_1, \p_1\p_3, \p_1\p_2, \p_1\p_{23})\\
%
  \geomtripp(\F_1,\F_2,\F_3)
&=   \geombir(\D_1, \p_1\p_{23}, \p_1\p_3, \p_1\p_2)
\end{array}
\;.\]
%
To simplify, we  denote from now on 
$\gZ_\indTriRap=\geomtri_\indTriRap(\F_1,\F_2,\F_3)\in\RR$.
%
In the algebraic case $\PP=\PP(\KK^3)$,  we have 
$\gZo =\log\abs{\Z}$,  
$\gZp =-\log\abs{1+\Z}$ and 
$\gZpp = \log\abs{1+\Z^{-1}}$
where $\Z\in\KK$ is the usual algebraic triple ratio $\Tri(\Tau)$
of the triple $\Tau$.
%
The geometric triple ratio $\gZ=\geomtri(\Tau)=(\gZo,\gZp,\gZpp)$ 
enjoys the following properties. 
It is invariant 
by cyclic permutations of the flags, and reversing the order
we get
%
 $\geomtri_1(\ov{\Tau})=-\geomtri_1(\Tau)$,
 $\geomtri_2(\ov{\Tau})=-\geomtri_3(\Tau)$.
%
We also have $\gZo+\gZp+\gZpp=0$, and 
the stronger following property :
%
for all $\indTriRap\in\ZZ/3\ZZ$, 
if $\geomtri_\indTriRap(\Tau)>0$ then $\gZ_{\indTriRap-1}(\Tau)=0$ and 
    $\gZ_{\indTriRap+1}(\Tau)=-\gZ_\indTriRap(\Tau)<0$.
%
Note that the three natural cases: 
$\gZ\in\RR_+(0,1,-1)$, 
$\gZ\in\RR_+(-1,0,1)$, and
$\gZ\in\RR_+(1,-1,0)$
%
subdivide in two types, as the case $\gZo=0$ is invariant under
reversing the order of $\Tau$, 
whereas the two other  cases are exchanged.    




%
We now turn to  the geometry in the interior
 of the Euclidean building $\Es$. 
%
A generic triple of ideal chambers $(\F_1,\F_2,\F_3)$ 
defines five natural flats in $\Es$:
the three flats $\Aij=\App(\F_\ii,\F_\jj)$
joining the opposite chamber $\F_\ii$ and $\F_\jj$,
%
the flat $\Ap=\App(\p_1,\p_2,\p_3)$ joining the  generic triple
of ideal singular points $(\p_1,\p_2,\p_3)$,
%
and the flat $\AD=\App(\D_1,\D_2,\D_3)$ joining $(\D_1,\D_2,\D_3)$.
%
We will show that there are also six particular points in $\Es$ naturally
associated with the configuration, that may be defined as the
orthogonal projections $\yTauE_\ii$ and $\yTauE_\ii^*$ (which happen
to be unique) of the boundary points
$\p_\i$ and $\D_\i$ on the flat $\Ajk$ 
where $\jj=\ii+1$ and $\kk=\ii+2$.
%

%
We say that $(\F_1,\F_2,\F_3)$ is of type {\em \typetripod{}}
if there exists a tripod in $\Es$  
joining the three (middle points of the) ideal 
chambers $(\F_1,\F_2,\F_3)$.  The set of centers of such
tripods is the intersection $I$ of the three flats $\Aij$.
%

%
We show that either 
the three flats $\Aij$ have nonempty intersection,
i.e. $(\F_1,\F_2,\F_3)$ is of type \typetripod,
or 
the two flats $\Ap$ and $\AD$ have non empty intersection $\Delta$,
which is then a {\em flat singular triangle} 
(that is, a triangle in $\Aa$ with singular sides)
%
(we then say that $(\F_1,\F_2,\F_3)$ is of type {\em  \typeflat{}}).
%
The two following results 
describe more precisely the two possible types,
and relate them with the points $\yTauE_\ii, \yTauE_\ii^*$ 
and the geometric triple ratio $\gZ$.
%
We denote by 
$\Cc=\{\vA\in\Aa/\ \vA_1 > \vA_2 >\vA_3\}$
the model Weyl chamber of $\Aa$ 
and we use the corresponding {\em simple roots coordinates} on $\Aa$,
that is $\vA=(\vA_1-\vA_2,\ \vA_2-\vA_3)$. 

%
\begin{theo}[Type \typetripod{}]
\label{theo- tripod}
The intersection $I=\App_{12} \cap \App_{23} \cap \App_{31}$ is
  nonempty if and only if 
$\gZo=   0$. 
%
Then $\gZp \geq 0$ and 
there exist a unique couple $(\vstE,\vstE^*)$ in $\Es$ such that
\begin{enumerate}
\item $\yTauE_1=\yTauE_2=\yTauE_3=\vstE$ 
and  $\yTauE_1^*=\yTauE_2^*=\yTauE_3^*=\vstE^*$ ;

\item $I$ is the segment $[\vstE,\vstE^*]$ ; 
\item $[\vstE,\vstE^*]$ is the unique shortest segment 
%
joining $\Ap$ to $\AD$.
\item 
\label{i- coords xx^*}
Identifying $\App_{\ii\jj}$ with $\Aa$ 
by a marked flat $\f:\Aa\mapsto \App_{\ii\jj}$ sending $\Cc$ to $\F_\jj$, 
in simple roots coordinates, we have
$\overrightarrow{\vstE\vstE^*}=(-\gZp,\gZp)$.
%
In particular $\vstE^*$ is on the ray $[\vstE,\p_\ij)$ 
from $\vstE$ to $\p_\ij$. 
%
\end{enumerate}
%
\end{theo}


\begin{figure}[h]
%
\includegraphics{figures/Es_triple_type_tripod.fig}
%

  \begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_Aij_tripod.fig}
%

{In $\Aij$.}
  \end{minipage}
%
\hfill
%
  \begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_Ap_tripod.fig}
    
In $\Ap$.
  \end{minipage}
%
\hfill
%
  \begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_AD_tripod.fig}
\vskip-2.5ex
{\hskip7em In $\AD$.}
  \end{minipage}


\caption{Type \typetripod}
%
\label{fig-  type tripod}
\end{figure}



%

\begin{theo}[Type \typeflat{}]
\label{theo- triangle}
%
The intersection $\Ap \cap \AD$ is nonempty if and only if
($\gZp=0$ or $\gZpp=0$), 
or, equivalently, if and only if  $\gZp\leq 0$. 
%
Then there exists a unique  flat singular triangle $\Delta$ with
  vertices $\vstE_1,\vstE_2,\vstE_3$
%
such that

\begin{enumerate}
\item $\Ap \cap \AD = \Delta$.
%

\item
\label{i- Aij cap Aik chamber tip x_\ii}
    $\Aij\cap\Aik$ is the Weyl chamber from $\vstE_\ii$ to $\F_\ii$ ;



\item
\label{i- xi xj}
Let $\ii\in\{1,2,3\}$ and $\jj=\ii+1$.
In a marked flat $\f:\Aa\mapsto \Aij$ sending $\Cc$ to $\F_\jj$, 
in simple roots coordinates, we have
$\overrightarrow{{\vstE_\ii}{\vstE_\jj}}
%
=(\posp{\gZo},\negp{\gZo})$
where $\posp{\gZo}=\max(\gZo,0)$ and $\negp{\gZo}=\max(-\gZo,0)$.
%
In particular $\vstE_\jj$ is on the ray from
$\vstE_\ii$ to $\p_\jj$ (if $\gZo\geq 0$) 
or $\D_\jj$ (if $\gZo\geq 0$).
 
 

\item 
\label{i- Delta opp Fi en xi}
 The germs of Weyl chambers at $\vstE_\ii$ 
respectively defined  by  $\Delta$ and $\F_\ii$ 
are opposite (in the spherical building of directions at $\vstE_\ii$).
%
In particular there exists a flat containing
  $\Delta$, and containing $\F_\ii$ in its boundary.
%
\end{enumerate}
%
Furthermore 
if $\gZo \geq 0$ we have
 $\vstE_\ii=\yTauE_{\ii-1}=\yTauE^*_{\ii+1}$ for all $\ii$,
%
and if $\gZo \leq 0$ we have $\vstE_\ii=\yTauE_{\ii+1}=\yTauE^*_{\ii-1}$ for
all $\ii$.
%
\end{theo}

The intersections of each flat with the four other flats form a
partition (i.e. a covering with disjoint interiors),
%
which is  described in figure
\ref{fig- type tripod} for the type \typetripod,  
and in figure \ref{fig-  type triangle in positive case}
for the type  \typeflat{}
%
(see Propositions \ref{prop- in Aij
  - intersections}, corollary \ref{coro- partition Aij} and \ref{prop-
  in Ap - intersections}).

\begin{figure}[h]
%
\includegraphics{figures/Es_triple_type_triangle.fig}
%


  \begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_Aij_triangle_pos.fig}
%

{In $\Aij$, with $\jj=\ii+1$.}
  \end{minipage}
%
\hfill
%
  \begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_Ap_triangle_pos.fig}
    
In $\Ap$.
  \end{minipage}
%
\hfill
%
\begin{minipage}[b]{0.3\linewidth}
\centering
%
\includegraphics{figures/Es_AD_triangle_pos.fig}
    
\vskip -2.5ex
{\hskip 5em In $\AD$.}
  \end{minipage}

\caption{Type \typeflat, in the case where $\gZo  \geq 0$
(the case $\gZo\leq 0$ is obtained from the case $\gZo\geq 0$
by reversing the order of the flags $\F_\ii$, 
i.e. by exchanging $1$ and $3$ and $\ii$ and $\jj$ in the above pictures).
}
\label{fig-  type  triangle in positive case}
\end{figure}
%


\medskip

%
The special case where hypotheses of both 
Theorems \ref{theo- tripod} and \ref{theo- triangle}
are satisfied correspond to the case where  $\gZo=\gZp=\gZpp=0$.
Then the five flats intersect in an unique point $\vstE$,
and, 
in the spherical building of directions at $\xE$,    
the triple of chambers induced by $\Tau=(\F_3,\F_2,\F_1)$ is generic.
%


%

%
In particular we recover the caracterization of \cite{Balser08} for
triples of points in $\bordinf\Es$ bounding a tripod in $\Es$. 
%
Note that M. Talbi  established some analogous  geometric classification
for interior triangles  in  discrete Euclidean buildings of type
$\typeA_2$ \cite{Tal01Th}.
%




%
Theorem \ref{theo- triangle} will be used in \cite{ParDegFG} to study
actions of punctured surface groups on Euclidean buildings of type
$\typeA_2$, using Fock-Goncharov parameters on ideal triangulations.
Theorem \ref{theo- triangle} enables us 
to associate to each triangle of the triangulation a flat singular
triangle in $\Es$, 
and, under simple hypotheses, connecting them  by gluing flat strips
between their edges, we obtain explicit nice invariant subcomplexes.
%
This allows to describe length spectra for large families of 
degenerations of convex projective structures
on surfaces.
%


%
We also show that generic quadruples in $\PP$
define a nice center in $\Es$, with various characterizations, 
%
see Proposition \ref{prop- center of a projective frame}
(this result generalizes to higer rank $\RR$-buildings of type
$\typeA_{\N-1}$). 
%



%


\paragraph{{\bfseries Aknowledgments}}
I would like to thank Frédéric Paulin 
for usefull discussions and comments. 
I also want to thank the members 
of the  Institut Fourier for their support.

triples_1_prelim.tex
%

%

\section{Preliminaries}
\label{s- prelim}

\subsection{The model flat 
\texorpdfstring{$(\Aa,\W)$}{(A,W)}
of type 
\texorpdfstring{$\typeA_{\N-1}$}{A\_N}
}
%
Let $\N\geq 2$ be an integer.  
%
The {\em model flat} of type $\typeA_{\N-1}$ is  the vector space 
$\Aa=\RR^\N/\RR(1,\ldots,1)$, endowed with the action of
the {\em Weyl group}
%
 $\W=\Sym_\N$ acting on $\Aa$ by permutation of
coordinates (finite reflection group). 
%
We denote by $\classA{\vRN}$
the projection in $\Aa$ of a vector $\vRN$ in $\RR^\N$.
%
The vector space $\Aa$ may be identified with the hyperplane 
$\{\vA=(\vA_1,\ldots,\vA_\N)\in\RR^\N/\ \sum_\indN
\vA_\indN=0\}$ of $\RR^\N$.
%
Recall that a vector in $\Aa$ is called {\em singular} if it belongs
to one the hyperplanes $\vA_\indN=\vA_\indNp$, and {\em regular} otherwise. 
%
A {\em (open) (vectorial) Weyl chamber} of $\Aa$ is a connected component of
regular vectors.
%
The {\em model Weyl chamber} 
is the simplicial cone  
$$\Cc=\{\vA\in\Aa/\ \vA_1 > \cdots >\vA_\N\}\;.$$
%
Its closure $\Cb$ is a strict fundamental domain for the action of
$\W$ on $\Aa$.
%
Recall that two nonzero vectors $\vA$ and $\vA'$ of $\Aa$ 
are called {\em opposite} if $\vA'=-\vA$.
Similarly, two Weyl chambers $\Ch$ and $\Chp$ of $\Aa$ 
are {\em opposite} if $\Chp=-\Ch$.
%
The {\em \typeC}  of a vector $\vA\in\Aa$ is its 
projection  (modulo
$\W$) in $\Cb$.

%

%
We denote by $\bord \Aa$ the sphere of unitary vectors in $\Aa$,
%
identified with the set $\PP^+(\Aa)=(\Aa-\{0\})/\RR_{>0}$ 
of rays issued from $0$,
%
and by $\bord: \Aa-\{0\} \to \bord \Aa$ the corresponding projection.
%
The {\em type (of direction)}  of 
a nonzero vector $\vA \in \Aa$ 
is its canonical projection 
%
in $\bord \Cb$.
%


We denote by
$(\eRRNo_1,\ldots,\eRRNo_\N)$ the canonical basis of $\RR^\N$.
%
\label{s- def type of vertices in bord Aa}
For $d=1,\ldots,\N-1$, we will say that  a nonzero vector in $\Aa$ (or a
point in the sphere $\bord\Aa$) is {\em singular of type $d$} if
its canonical projection in $\bord\Cb$ is
$\classA{\eRRNo_1+\cdots+\eRRNo_d}$.
%


%


%

%
The  {\em simple roots} (associated with $\Cc$) are the following linear forms on $\Aa$
$$\rac{\indN} : \vA\mapsto +\vA_\indN-\vA_{\indN+1}$$
for $\indN=1,\ldots,\N-1$.
%
The set of simple roots is denoted by $\RacSimples$.
%
We will also use the root 
$\rac\N : \vA \mapsto\vA_\N-\vA_1 $
%
satisfying
$$\rac1+\cdots+\rac\N =0\;.$$
%

\begin{figure}[h]
  \includegraphics[scale=1]{figures/Aa.fig} 
  \caption{The model flat $\Aa$ of type
    $\typeA_2$ ($\N=3$).}
  \label{fig- Aa}
\end{figure}



%
The vector space $\Aa$ is endowed with 
the unique $\W$-invariant Euclidean scalar product, 
which is  well defined up to homothety (induced by the
standard Euclidean scalar product of $\RR^\N$).
%
We will normalize it 
%
by requiring that the simple roots have unit norm, 
i.e. the distance between the two hyperplanes 
with equation $\rac\indN=0$ and $\rac\indN=1$
is $1$ for one (all) $\indN$. 
%
When $\dim\Aa=1$, 
%
we will  identify $\Aa$ 
%
with $\RR$ by the basis $\{\classA{\eRRNo_1}\}$, i.e. 
by the map $\sRR \mapsto \sRR \classAcoordsll{1}{0}$ (which is an
isometry in the above normalization).
%






\subsection{Projective spaces}
\label{ss- projective spaces} 
%
We here collect notations and definitions for projective spaces, which
will be used throughout this article.
%
Let $\PP$ be projective space  of dimension $\N-1$, with $\N\geq 2$.
%
We denote by $\AllFlags(\PP)$ the set of flags of $\PP$,
that is  increasing sequences $(\V_1,\ldots,\V_{\M})$ 
of proper linear subspaces of $\PP$.
%
We denote by $\PP^*$  the set  of hyperplanes in $\PP$ 
(dual projective space).
%
Two maximal flags are called {\em opposite} if they are in generic position.

%
A finite subset $\p_1, \ldots, \p_\M$ in $\PP$,  with $2\leq \M \leq \N$, 
is {\em generic} if it is not contained in any linear subspace of
dimension $\M-2$ of $\PP$. 
%
Then it is contained in a unique $(\M-1)$-dimensional linear subspace of
$\PP$, which will be denoted by $\p_1\oplus \cdots \oplus \p_{\M}$.
%
When $\M=2$, we will also denote $\p\oplus\q$ by $\p\q$.
%
A {\em frame}
%
of $\PP$ 
is a generic $\N$-tuple. 
%
A {\em projective frame} in $\PP$
%
is a $(\N+1)$-tuple $(\p_0,\p_1, \ldots,\p_\N)$  
of points in $\PP$
%
in generic position, 
i.e. such that  
$(\p_0,\ldots, \widehat{\p_\indN},\ldots, \p_\N)$ is a frame in $\PP$
(generic $\N$-tuple) for all $\indN$.


%
If $\p$ is a point in $\PP$, 
we denote by $\PP/\p$
%
the  set  of lines through $\p$,
which is a projective space of dimension $\N-2$ for the induced
structure, 
and $\projP_\p:\q \mapsto \p\q$ the canonical projection
from $\PP-\{\p\}$ to $\PP/\p$ 
(which is a morphism of projective spaces).
%
If $\p$ is a point of $\PP$ and $\H$ an hyperplane with $\p\notin \H$,
 then the projection $\projP_\p$ induces a canonical 
isomorphism  $\projP_{\H\p}:\H\isomto \PP/\p$ (called {\em perspectivity}).
%


Note that if $(\p_1,\ldots, \p_\M)$ is generic in $\PP$, 
then its projection  
$\projP_{\p_1}(\Frame)=(\p_1\p_2,\ldots, \p_1\p_\M)$  
at $\p_1$
is generic in  $\PP/\p_1$ 
(in particular the projection of a (projective) frame at
one of its points is a (projective) frame).




\subsection{Spherical buildings of type 
\texorpdfstring{$\typeA_{\N-1}$}{A\_{N-1}}
 and associated projective spaces}
\label{ss- proj space ass to a spherical An building}
%
See \cite[\S 6]{Tits74}.
%
A spherical building $\B$ of type $\typeA_{\N-1}$ 
%
is the building of flags 
of an associated projective space $\PP=\PP(\B)$ of dimension $\N-1$. 
%
For $d=0,1,\ldots,\N-1$, 
the set of linear subspaces of dimension $d$ of $\PP$ 
identifies with the subset of vertices of type $d+1$
%
 of $\B$.
%
In particular, the projective space  $\PP$ itself
is identified with the set of vertices of type $1$ of $\B$.
%
Note that the set of vertices of type $\N-1$ is then identified with
the dual projective space $\PP^*$.
%


%
In the algebraic case, 
that is when $\B$ is the spherical building of flags 
of
%
some vector space $\V$ of dimension $\N$ over a field $\KK$, 
then $\PP=\PV$.

%
A basic fact is that 
the frames $\Frame$  in $\PP$ correspond to the apartments of $\B$ 
by $\Frame \mapsto \AllFlags(\Frame)$. 
%



%
 
%
Recall that, in (the geometric realization of
modelled on $(\bord\Aa,\W)$ of) 
a spherical building, any two points (resp. chambers)
are contained in a common apartment,
%
and that they are  {\em opposite} if they
are opposite in that apartment, 
that is, 
for two points $\xi,\xi'$, 
if and only if $\angle(\xi,\xi')=\pi$ 
for the canonical metric $\angle$ on $\B$.
%
Note that $\p\in\PP$ and $\H\in\PP^*$ 
are opposite if and only if 
$\angle(\p,\H)=\pi$, 
if and only if $\p\notin H$.
%
Two chambers are opposite if and only if they are opposite
as maximal flags in $\PP$.
%
In particular, in the type $\typeA_2$ case, two chambers 
$\F_1=(\p_1,\D_1)$, $\F_1=(\p_2,\D_2)$ are opposite 
if and only if $\p_1\notin \D_2$ and $\p_2 \notin \D_1$.




%

%
For any simplex $\sigma$ of $\B$ the {\em residue}  $\Star(\sigma)$ 
of $\sigma$ is  the
spherical building formed by 
the simplices of $\B$ containing $\sigma$. 
%
If $\H$ is a hyperplane of $\PP$, 
the residue $\Star(\H)$ of $\H$ in $\B$ 
is the subset of  flags of $\PP$ containing $\H$
and it canonically identifies 
with the spherical building $\AllFlags(\H)$ of flags of the $\N-1$
dimensional projective space $\H$ 
by the map $(\V_1,\ldots,\V_{\M},\H)\mapsto (\V_1,\ldots,\V_{\M})$.
%
The residue $\Star(\p)$ of a point $\p$ in $\PP$, 
i.e. the set of  flags of $\PP$ containing $\p$,
identifies canonically
with the flag building $\AllFlags(\PP/\p)$ of $\PP/\p$ 
by the map 
%
$(\V_1=\p,\ldots,\V_{\M})\mapsto (\V_2/\p,\ldots,\V_{\M}/\p)$.
%
If $\p\notin \H$ then the projection $\projP_\p$ induces a canonical 
isomorphism (of spherical buildings) $\projP_{\H\p}:\Star(\H)\isomto
\Star(\p)$ (perspectivity).
%







\subsection{Euclidean buildings}
We refer for example to \cite{ParImm} for the definition and
properties of (real) Euclidean buildings we use below 
(see also \cite{Tits86}, \cite{KlLe97}, \cite{Rousseau09}). 
%
From now on,  $\Es$ will denote 
a (not necessarily discrete) Euclidean building of type
$\typeA_{\N-1}$.
%
Recall that 
$\Es$ is a $\CAT(0)$ metric space 
%
endowed with a (maximal) collection $\Apps$ of
isometric embeddings $\f:\Aa \to \Es$ called  {\em marked apartments},
or {\em marked flats} by analogy with  Riemannian symmetric spaces,
satisfying the following properties
%
\begin{mydescription}
  \item[{\bfseries (A1)}] $\Apps$ is invariant by precomposition by $\Waff$
    ;
  \item[{\bfseries (A2)}] If $\f$ and $\f'$ are two marked flats, then the
    transition map $\f^{-1}\circ \f'$ is 
%
in
 $\Waff$
    ;
  %
  \item[{\bfseries (A3')}] Any two rays of $\Es$ are initially 
 contained in a common marked flat.
\end{mydescription}
 The {\em flats} 
%
(resp. the {\em Weyl chambers}) 
of $\Es$ are the images of $\Aa$ (resp. of $\Cc$)
%
by the marked flats.




%
\subsubsection*{Algebraic case}
\label{s- Es(V)}
%
Let $\KK$ be an ultrametric field, i.e. a field endowed with 
an ultrametric absolute value $\abs{\cdot}$ (not necessarily
discrete). 
%
When $\V$ is a finite $\N$-dimensional vector space over $\KK$, 
we denote by  $\Es=\Es(\V)$ the Euclidean building associated with
$\G=\PGL(\V)$.
%
 We refer for example to \cite{ParImm} 
for the model of norms for $\Es$ (see \cite{GoIw63}, \cite{BrTi84}).
%
To each  basis $\eVb$ of $\V$ is then
associated a marked flat $\f_\eVb:\Aa\to \App_\eVb$,
%
such that, 
if $\aG$ is an element of $\G$ with diagonal
matrix $\diag(\aK_1,\ldots,\aK_\N)$ in the basis $\eVb$, 
%
then $\aG$ translates the flat $\App_\eVb$  by the  vector
$$\nu(\aG)=\classA{(\log\abs{\aK_\indN})_\indN}$$
in $\Aa$ (identifying the flat $\App_\eVb$ with the model flat
$\Aa$ through the marking $\f_\eVb$).
%

\subsection{Spherical building and projective space at infinity}
%
The CAT(0) boundary $\bordinf\Es$ of $\Es$ is the geometric
realization modeled on $(\bord\Aa,\W)$ of a spherical building of type
$\typeA_{\N-1}$ (whose chambers are the boundaries of the Weyl
chambers of $\Es$).
%
It will be identified with the building of flags on the associated
projective space $\PP=\PP_\infty(\Es)$ 
(whose points are the vertices of type $1$ of $\bordinf \Es$). 
%
If $\chp$ and $\chm$ are opposite ideal chambers, then
we denote by 
$\App(\chm,\chp)$ the unique flat joining $\chm$ to $\chp$ in $\Es$.
%
If $\Frame$ is a frame of $\PP$, then 
there is a unique flat  $\App(\Frame)$ containing $\Frame$ in its boundary.
%










\subsection{Local spherical building and projective space at a point}
%
Recall that, in Euclidean buildings, 
two (unit speed) geodesic segments issued from
a common point $\xE$ have zero angle 
%
if and only if 
they have same germ at $\xE$ (i.e. coincide in a neighborhood of $\xE$).
%
A {\em direction at $\xE\in\Es$} is 
a germ of nontrivial geodesic segment from $\xE$.
%
A direction, geodesic segment, ray or line 
has a well-defined {\em type (of direction)} in $\bord\Cb$, 
which is its canonical
projection (through a marked flat) in $\bord\Cb$.
 %
 %
 %
It is called {\em singular} or {\em regular} accordingly.

%
The {\em space of directions} 
%
at  $\xE$ of $\Es$ 
is the quotient space of non trivial geodesic segments from $\xE$ for
this relation, with the induced angular metric,
and is denoted by  $\TangS_\xE \Es$. 
%
We denote by   $\TangS_\xE:\Es-\{\xE\}\to \TangS_\xE \Es$,
$\yE \to \TangS_\xE\yE$, 
the associated projection.
Its extension to the boundary at infinity 
will also be denoted by $\TangS_\xE: \bordinf\Es \to \TangS_\xE \Es$,
$\xi \to \TangS_\xE\xi$
and called the {\em canonical projection}.

%
The space of directions $\TangS_\xE \Es$ inherits the structure 
of a spherical $\typeA_{\N-1}$-building,
whose apartment are the germs $\TangS_\xE \App$ at $\xE$
of the flats $\App$ of $\Es$ passing through $\xE$,
%
and whose chambers are the germs $\TangS_\xE \Ch$ at $\xE$ 
of the Weyl chambers $\Ch$ of $\Es$ with vertex $\xE$ 
(see for example \cite{ParImm}).
%
The canonical projection $\TangS_\xE: \bordinf\Es \to \TangS_\xE \Es$
sends chambers to chambers 
(and, more generally, simplices to simplices)
and  
preserves the type (in $\bord\Cb$) of points. 

%
The {\em local projective space} $\PP_\xE = \PP_\xE(\Es)$ at $\xE$ 
is the projective space of dimension $\N-1$ associated with 
 the spherical building $\TangS_\xE \Es$ of type $\typeA_{\N-1}$
(see \S \ref{ss- proj space ass to a spherical An building}).
Its underlying set is the set  of vertices of type $1$ of $\TangS_\xE\Es$.

%
The canonical projection $\TangS_\xE: \bordinf\Es \to \TangS_\xE \Es$
induces  (by restriction to vertices) 
a surjective morphism (of projective spaces)  
$\TangS_\xE:\PP \to \PP_\xE$
%
from the projective space at infinity $\PP$ 
to the local projective space $\PP_\xE$ at $\xE$.
%
Note that, in particular, if $\Frame$ is a frame of $\PP$, then 
%
\label{s- x is in App(Frame) iff TangS_xFrame is a frame}
%
$\xE$ belongs to the associated flat $\App(\Frame)$ 
if and only if 
$\TangS_\xE(\Frame)$ is a frame of $\PP_\xE$. 





\subsection{Transverse spaces at infinity}
\label{s- transverse spaces at infinity}
%
(See for example \cite[\S 8]{Tits86}, 
\cite[1.2.3]{Leeb00}, \cite[\S 4]{MSVM14-I}.)
%
Let $\xi$ be a vertex of $\bordinf\Es$ of type $1$ or $\N-1$,
i.e. either a point $\p$ in the projective plane at infinity $\PP$ or
a hyperplane $\H$ of $\PP$. 

%
The {\em transverse space} $\Es_\xi$ at $\xi$
%
may be defined, from the metric viewpoint 
(as in \cite[1.2.3]{Leeb00}),
as 
%
the quotient space of the set of all rays to $\xi$  by the
pseudodistance $\dxi$ given by
$$\dxi(\ray_1,\ray_2)=\inf_{\tRR_1,\tRR_2}\deucl( \ray_1(\tRR_1),
\ray_2(\tRR_2) ) \;.$$%
%
We denote by  $\projE_\xi:\Es\to\Es_\xi$ the canonical projection 
%
(which maps $\xE$ to the class of the unique ray from $\xE$ to $\xi$).
%
The space $\Es_\xi$  is a Euclidean building 
%
of type $\typeA_{\N-2}$,
%
whoses flats are the projections to $\Es_\xi$ of the flats
of $\Es$ passing by $\xi$.
%

%


%
In the algebraic case, i.e. when $\Es=\Es(\V)$, 
the transverse space $\Es_\H$ canonically
identifies with the building $\Es(\H)$ of $\H$, 
where $\H$ is seen as  an hyperplane of $\V$,
%
and $\Es_\p$ identifies with $\Es(\V/\p)$, 
where $\p$ is seen as a $1$-dimensional subspace of $\V$.
%

%
The spherical building $\bordinf \Es_\xi$ at infinity  of $\Es_\xi$ 
identifies canonically with the residue $\Star(\xi)$ of $\xi$.
%
In particular,  if  $\p$ is a point in $\PP$, 
the projective space at infinity of $\Es_\p$ 
%
identifies with  $\PP/\p$, 
%
and if $\H$ is an hyperplane of $\PP$, 
the projective space at infinity of $\Es_\H$ identifies with $\H$.

%
\label{projection of a flat and frames}
%
If $\Frame=(\p_1, \ldots, \p_\N)$  is a frame in $\PP\subset\bordinfEs$, 
then the projection on $\Es_{\p_1}$ of the flat
$\App(\p_1, \ldots, \p_\N)$ 
is the flat defined by the projection
$\projP_{\p_1}(\Frame)=(\p_1\p_2,\ldots, \p_1\p_\N)$ of the frame
$\F$, i.e. $\projE_{\p_1}(\App(\Frame))=\App(\projP_{\p_1}(\Frame))$.
%



%
We now describe the canonical isomorphism 
$\projE_\ximp:\Es_\xim \isomto \Es_\xip$ 
for  opposite points $\xim,\xip$ of $\bordinfEs$.
%
The union  $\Esximp$ of all geodesics joining $\xim$ to $\xim$ 
is a convex closed subspace and a subbuilding.
%
We denote by $\Esximp=\Es^\ximp\times \RR$
the canonical decomposition.
%
The restriction of the projection $\projE_\xip$ to $\Esximp$ is surjective
and factorize through the projection on the first factor, 
inducing a canonical isomorphism of Euclidean buildings
$\Es^\ximp \isomto \Es_\xip$.
We similarily have a canonical isomorphism
$\Es^\ximp \isomto \Es_\xim$,
%
so it induces a canonical isomorphism 
$\projE_\ximp:\Es_\xim \isomto \Es_\xip$.
%
\label{prel- persectivities comes from an 
isomorphism of the transverse trees}%
%
It is easy to see that the map $\projE_\ximp$ 
extends to the boundaries at infinity of 
$\Es_\xim$ and $\Es_\xip$
by the canonical isomorphism of spherical buildings 
$\projP_\ximp:\Star(\xim) \isomto \Star(\xip)$ (perspectivity).
%













\subsection{The 
\texorpdfstring{$\Aa$}{A}-valued 
Busemann cocycle}
%
Let $\ch$ be a chamber at infinity of $\Es$. 
%
We may define the $\Aa$-valued {\em Busemann cocycle} 
$\Buso_\ch:\Es\times\Es\to\Aa$
 by the property
$$\Bus{\ch}{\f(\vA)}{\f'(\vA')}=\vA'-\vA$$
for all marked flats $\f,\f':\Aa\to \Es$ sending $\bord\Cc$ to $\ch$
and {\em very strongly asymptotic}
%
that is 
such that $\deucl(\f(\ray(\tRR)), \f'(\ray(\tRR)))$ goes to zero when $\tRR\tend
\pinfty$ for one (all) regular ray $\ray$ in $\Cc$
(which in Euclidean buildings is equivalent to: 
$\f=\f'$ on some subchamber $\vA"+\Cc$).
%
We clearly
%
have
$$\Bus{\ch}{\xE}{\zE}=\Bus{\ch}{\xE}{\yE}+\Bus{\ch}{\yE}{\zE}\;.$$
%
When $\dim\Aa=1$, it coincides with the usual Busemann cocycle, 
\label{ss- usual Bus cocycle}%
%
which is defined for $\xi \in \bordinf\Es$ by
%

%
$$\Bus{\xi}{\xE}{\yE}=\lim_{\zE\ra \xi}\deucl(\xE,\zE)-\deucl(\yE,\zE)\;.$$
%



%
In type $\typeA_2$ case, the simple root coordinates of
$\Aa$-valued Busemann cocycles
 may be determined by projecting in transverse trees at infinity, 
using the following relations (using the normalization of the metric).
%
\begin{equation}
  \label{eq- projections and Busemann cocycle}
  %
  %
  \begin{array}{rl}
%
    \rac1(\Bus{(\p,\D)}{\xE}{\yE})
    & = \Bus{\p} {\projE_\D(\xE)} {\projE_\D(\yE)}\\
%
    \rac2(\Bus{(\p,\D)}{\xE}{\yE})
    & = \Bus{\D} {\projE_\p(\xE)} {\projE_\p(\yE)} \;.
%
  \end{array}
\end{equation}
%


\bigskip


%

We now turn to cross ratios.
%




\subsection{Cross ratio on the boundary of a tree}
\label{ss- geometric cross ratio on the boundary of a tree}
(See \cite[\S 7]{Tits86}, 
and for a more general setting \cite{Otal92}, \cite{Bourdon96}.)
%
In this section, we suppose that $\Es$ is 
%
a (metric) $\RR$-tree.
%
Given three distinct ideal points $\xi_1,\xi_2,\xi_3$ in $\bordinfEs$,
we denote by $\centre(\xi_1,\xi_2,\xi_3)$ 
the {\em center} of the ideal triple $\xi_1,\xi_2,\xi_3$, 
that is the unique common intersection point 
of the three geodesic lines joining two of the three points.
%
Note that $\centre(\xi_1,\xi_2,\xi_3)$ is the (orthogonal)
projection of $\xi_3$ on the geodesic joining $\xi_1$ to $\xi_2$.
%
We denote by $\Bus{\xi}{\xE}{\yE}$ the Busemann cocycle 
(see \S \ref{ss- usual Bus cocycle}).
%


%
\noindent {\advance\linewidth by-5cm
\begin{minipage}{\linewidth}
Define the {\em cross ratio} of four pairwise distinct 
points $\xi_1$, $\xi_2$, $\xi_3$, $\xi_4$ in  $\bordinfEs$ 
%
by
%
$$ \geombir(\xi_1, \xi_2, \xi_3,\xi_4)
=\frac{1}{2}(\ell_{12}-\ell_{23}+\ell_{34}-\ell_{41})$$
%
where $\ell_\ij$ is the length of the geodesic in $\Es$ 
from $\xi_\indQuadP$ to $\xi_\indQuadPp$ after removing disjoint
fixed horoballs centered at each $\xi_\indQuadPpp$.
\end{minipage}
}
%
\hfill
%
\begin{minipage}{4cm}
%
\centering\includegraphics[scale=0.7]{figures/Tree_geom_crossratio.fig}
%
\end{minipage}

%


%

%
The cross ratio naturally extends to 
non generic quadruples
that are {\em \nondegeneratedQ{}}, 
%
that is 
quadruples $(\xi_1,\xi_2,\xi_3,\xi_4)$ 
{\em without triple point}  
(i.e. any three of the points are not equal),
which is equivalent to the following condition:
\begin{equation}
\label{eq- nondegeneratedQ}
  (\xi_1 \neq \xi_4 \mbox{ and } \xi_2 \neq \xi_3)
  \mbox{ or }
  (\xi_1 \neq \xi_2 \mbox{ and }  \xi_3 \neq \xi_4)
\;.
\end{equation}
%
  
We then set
$$
\geombir(\xi_1,\xi_2,\xi_3,\xi_4) 
= \left\{
\begin{array}{cl}
0       & \mbox{ when } \xi_1 = \xi_3 \mbox{ or } \xi_2 = \xi_4\\
\minfty & \mbox{ when } \xi_1 = \xi_2 \mbox{ or } \xi_3 = \xi_4\\
\pinfty & \mbox{ when } \xi_1 = \xi_4 \mbox{ or } \xi_2 = \xi_3
\end{array}
\right.%
\;.
$$
%


%
We now recall some basic properties that we will use.

The cross ratio may be read inside the tree on the oriented geodesic
from $\xi_3$ to $\xi_1$, as the oriented distance 
$\ovra{\xT\yT}$ 

%
\noindent {\advance\linewidth by-5cm
\begin{minipage}{\linewidth}
from the center $\xT$  
of the ideal triple
$\xi_3,\xi_1,\xi_2$ to the center $\yT$
of the ideal triple $\xi_3,\xi_1,\xi_4$:
%
\begin{equation}
\label{eq- geombir and centers of tripods}
  \geombir(\xi_1, \xi_2, \xi_3,\xi_4)
  =\ovra{\xT\yT}=\Bus{\xi_1}{\xT}{\yT}
\;.
\end{equation}
\end{minipage}
}
%
\hfill
%
\begin{minipage}{5cm}
%
\centering\includegraphics{figures/Tree_geom_crossratio_centers_tripods.fig}
%
\end{minipage}


%

\medskip


%
The cocycle identity 
%
is
$$\geombir(\xi_1, \xi_2, \xi_3,\xi_4)+\geombir(\xi_1, \xi_4,\xi_3,\xi_5)=
\geombir(\xi_1, \xi_2, \xi_3,\xi_5)\;.$$
%

%
The cross ratio $\geombir$ is left unchanged by the double
transpositions
%
\label{ss- geombir (12)(34)}  
%
and changed to $-\geombir$ by
$(13)$ and $(24)$.
%
We now consider $3$-cyclic permutations of the three last terms. 
%
We have
%
\begin{equation}
  \label{eq- sum of 3-cyclic perm of geom bir is 0} 
    \geombir(\xi_1, \xi_2, \xi_3, \xi_4)
  + \geombir(\xi_1, \xi_4, \xi_2, \xi_3)
  + \geombir(\xi_1, \xi_3, \xi_4, \xi_2) = 0
  \;.
\end{equation}
%
Moreover, the following     
{\em ultrametricity} property (specific to the case of trees) 
is easy to prove using \eqref{eq- geombir and centers of
  tripods} (see \cite[\S 7, prop. 3]{Tits86}):
\begin{equation}
\label{eq- geombir is ultram}
\begin{array}{ll}
\mbox{If } \geombir(\xi_1, \xi_2, \xi_3,\xi_4)>0, 
&\mbox{then } \geombir(\xi_1, \xi_3, \xi_4,\xi_2)=0\\
&\mbox{and }
\geombir(\xi_1, \xi_4, \xi_2,\xi_3)=-\geombir(\xi_1,
\xi_2,\xi_3,\xi_4)
\end{array}
\;.
\end{equation}
%
Note that \eqref{eq- geombir is ultram}
is equivalent 
(under \eqref{eq- sum of 3-cyclic perm of geom bir is  0}) 
to
%
\begin{equation}
\label{eq- geombir is ultram via max}
\geombir(\xi_1, \xi_2, \xi_3,\xi_4) 
\leq \max(0,-\geombir(\xi_1, \xi_4,  \xi_2,\xi_3))%
\;.\end{equation}
%
which in the algebraic case 
follows from the symmetry properties of the cross ratio 
under $3$-cyclic permutations \eqref{eq- sym bir 1342}. 
%










\subsection{Algebraic case: link with usual cross ratio}
Suppose that $\Es$ is the tree 
$\Es(\V)$ associated with  a 2-dimensional
vector space $\V$ over an ultrametric field $\KK$ 
(see Section \ref{s- Es(V)}).
%
Then $\bordinfEs$ identifies with the projective line $\PV$.

The usual cross ratio $\Bir$ on $\PV$ of a 
\nondegeneratedQ{} quadruple of points 
(see \eqref{eq- nondegeneratedQ}) 
is defined by (following the convention of \cite{FoGoSPC},
and taking values in  $\KK\cup\{\infty\}$)
\begin{equation}
\label{eq- def cross ratio}  
\Bir(\aK_1,\aK_2,\aK_3,\aK_4)
=\frac{(\aK_1 -\aK_2 )(\aK_3 -\aK_4 )}
      {(\aK_1 -\aK_4 )(\aK_2 -\aK_3 )}
\end{equation}
in any affine chart $\PV\isomto \KK\cup\{\infty\}$,  
so that $\Bir(\infty,-1,0,\aK)=\aK$.


The  cross ratio $\geombir$ 
defined in section \ref{ss- geometric cross ratio on the boundary of a tree}
will then be called the {\em geometric} cross ratio, to distinguish it
from $\Bir$, which will be called the  {\em  algebraic} cross ratio. 
%
They are  then related  as follows:
%
\begin{equation}
  \label{eq- link geom and alg cross ratio} 
 \geombir(\xi_1, \xi_2, \xi_3,\xi_4)
=\log\abs{\Bir(\xi_1, \xi_2,  \xi_3,\xi_4)}
\;.\end{equation}
%
\begin{proof}
%
Let $\xT_4=\centre(\xi_3,\xi_1,\xi_2)$ and
$\xT_2=\centre(\xi_3,\xi_1,\xi_4)$.
%
In a suitable basis $\eVb=(\eV_1,\eV_2)$ of $\V$, 
we have in homogeneous coordinates
  $\xi_1=[1:0]$, $\xi_3=[0:1]$, $\xi_2=[-1:1]$
 and $\xi_4=[\bK:1]$, 
where $\bK=\Bir(\xi_1, \xi_2,\xi_3,\xi_4)$. 
%
Then $\g=
  \begin{pmatrix}
-\bK&0\\
0&1
  \end{pmatrix}$
fixes $\xi_1$ and $\xi_3$ and sends $\xi_2$ to $\xi_4$.
Hence $\g(\xT_4)=\xT_2$. In the flat $\App(\xi_3,\xi_1)$ identified with
$\Aa=\RR^2/\RR(1,1)$ by the marked flat $\f_\eVb$,
%
 we have
$\ovra{\xT_4\xT_2}=\nu(\g)
=\classA{(\log\abs{\bK},0)}$, 
hence $\ovra{\xT_4\xT_2}=\log\abs{\bK}$ as needed.
%
\end{proof}
%




We recall that the algebraic cross ratio $\Bir$
satisfies the following symmetry properties:
%
It is left unchanged by the double transpositions
%
and changed to $\Bir^{-1}$ by  $(13)$ and $(24)$. 
%
Furthermore we have 
an additional symmetry under $3$-cycles 
not satisfied by the geometric cross ratio:
%
\begin{equation}
\label{eq- sym bir 1342}
%
    \begin{array}{rl}
      \Bir(\aK_1,\aK_3,\aK_4,\aK_2)&=-1-\Bir(\aK_1,\aK_2,\aK_3,\aK_4)^{-1}\\
      \Bir(\aK_1,\aK_4,\aK_2,\aK_3)&=-(1+\Bir(\aK_1,\aK_2,\aK_3,\aK_4))^{-1}  
    \end{array}
\;.\end{equation}
%





\subsection{Cross ratio on the boundary 
  of an \texorpdfstring{$\typeA_2$}{A\_2}-Euclidean building}
\label{ss- geometric cross ratio on a projective plane}
See \cite{Tits86}. 
%

%
Let $\Es$ be a Euclidean building of type $\typeA_2$, and $\PP$ the
associated projective plane at infinity. 

%
Let $(\p_1,\p_2,\p_3,\p_4)$ be a \nondegeneratedQ{} quadruple
of points of $\PP$ on a common line $\D$. 
Then their {\em cross ratio} $\geombir(\p_1,\p_2,\p_3,\p_4)$
(i.e. {\em projective valuation} in \cite{Tits86})
is by definition 
their cross ratio as ideal points of the transverse tree $\Es_\D$.
%
The cross ratio of a \nondegeneratedQ{} quadruple of lines in $\PP$ 
passing through a common point $\p$ 
is similarly defined 
as their cross ratio 
as ideal points of the transverse tree $\Es_\D$.

The main new property is that perspectivities preserve cross ratio:

\begin{prop}
Let $\p$ is a point of $\PP$ 
and $\D$ is a line of $\PP$ 
with $\p \notin \D$
the canonical isomorphisms (perspectivities)
$\projP_{\p\D}: \Star(\D)\isomto \Star(\p)$, $\q \mapsto \p\q$ 
and 
$\projP_{\D\p}: \Star(\p)\isomto \Star(\D)$, $\L \mapsto \D\cap \L$,
preserve the cross ratio $\geombir$, i.e.
\begin{equation}
\geombir(\p_1, \p_2,\p_3, \p_4)=\geombir(\q\p_1, \q\p_2,\q\p_3, \q\p_4)  
\end{equation}
\begin{equation}
\geombir(\D_1, \D_2,\D_3, \D_4)
=\geombir(\L\cap\D_1,\L\cap\D_2,\L\cap\D_3, \L\cap\D_4)  
\end{equation}
\end{prop}
%

%
\begin{proof}
The perspectivity $\projP_{\p\D}$ comes from 
the canonical isometry  
$\projE_{\p\D} : \Es_\D \to \Es_\p$
between the associated transverse trees
(see \ref{prel- persectivities comes from an 
isomorphism of the transverse trees}),
%
which preserves the centers of ideal triples, 
i.e.  for  all pairwise distinct $p_1, \p_2, \p_3$ in $\D$ 
we have
\begin{equation}
\projE_{\p\D} ( \centre(\p_1, \p_2, \p_3) )
= \centre(\p\p_1, \p\p_2, \p\p_3)
\;.\end{equation}
%
It follows that $\projP_{\p\D}$  preserves cross ratios.

Similarly,
for  all pairwise distinct lines $\L_1, \L_2, \L_3$ through $\p$ 
we have
\begin{equation}
\label{eq- projE_Dp preserves centers}
\projE_{\D\p} ( \centre(\L_1, \L_2, \L_3) )
= \centre(\D\cap\L_1, \D\cap\L_2, \D\cap\L_3)
\; ,\end{equation}
so $\projP_{\D\p}$  preserves cross ratios.
%
\end{proof}



triples_2_basic_ideal_configs.tex
%




\section{Some basic ideal configurations}


\subsection{
  Extension of orthogonal projection 
  to the boundary 
  in  general CAT(0) spaces
}
%
In this section, we study orthogonal projections of ideal points on
a convex subset 
in general CAT(0) spaces. 
%
More precisely, we  will need the following basic property:
the usual orthogonal projection on a proper convex subset $\Y$ 
 extends to the boundary outside 
the closed $\frac{\pi}{2}$-neighborhood of $\bordinf\Y$ for the Tits metric
(note that the projection is no longer unique).
%
This property is quite elementary 
but we did not see it 
in the classical litterature, 
so we include the proof.
%
We refer to the standard reference book \cite{BrHa99} for CAT(0) spaces.
%
We denote by $\angleTits(\xi,\eta)$ the Tits angle between two ideal points.
%
For $A\subset \bordinf\Es$,
let $\angleTits(\xi,A)=\inf_{\eta\in A}\angleTits(\xi,\eta)$.

\begin{prop}
  \label{prop- CAT0 - projection of ideal points}
  %
  %
  %
  Let $\Y$ be a convex subspace of a CAT(0) space
  $\Es$ which is proper for the induced metric, 
  and $\xi$ a point in $\bordinf\Es$. 
  %
  %
  Suppose that $\angleTits(\xi,\bordinf\Y)> \frac{\pi}{2}$. 
  %
  Then there exists $\xE\in\Y$
  such that $\xE$ is a (orthogonal) projection of $\xi$ on $\Y$, i.e.
  $\angle_\xE (\xi, \yE) \geq \frac{\pi}{2}$ for all $\yE\in \Y$.
\end{prop}

\begin{proof}
  Consider a sequence $\xE_\n\to \xi$ in $\Es$, and let $\yE_\n$ be the
  projection of $\xE_\n$ on $\Y$.
  If $(\yE_\n)_{\n\in\NN}$ is not bounded then up to passing to a subsequence
  $\yE_\n\to \eta$ in $\bordinf\Y$. 
  Then for any fixed $\yE$ in $\Y$ we have  
  $\angle_\yE(\xi,\yE_\n) \leq \frac{\pi}{2}$ for all $\n$,
  hence $\angle_\yE(\xi,\eta) \leq \frac{\pi}{2}$.
  Therefore  $\angleTits(\xi,\eta) \leq \frac{\pi}{2}$.
  %
  Thus $(\yE_\n)_{\n\in\NN}$ is bounded, hence, since   $\Y$ is proper, 
  it has a converging subsequence with limit
  point $\xE$.
\end{proof}



\subsection{Centers of generic 
\texorpdfstring{$(\N+1)$}{(\N+1)}-tuples.}
%
In this section, we show that the notion of center of ideal triples
in trees extends in Euclidean buildings of type $\typeA_{\N-1}$, for
generic $(\N+1)$-tuples of points (or hyperplanes) in the associated
projective space at infinity (Proposition \ref{prop- center of a projective frame}).
%

%
Let $\Es$ be a Euclidean building of type $\typeA_{\N-1}$, 
and  $\PP$ be its projective space at infinity  
(i.e., the set of singular points of type $\stypep$ in $\bordinf\Es$,
see section \ref{s- prelim}).  
%
Recall (see section \ref{ss- projective spaces}) that 
a {\em projective frame} 
in a projective space of dimension $\N-1$ 
is  a generic $(\N+1)$-tuple of points.

We first observe that we have
the (orthogonal) projection of a point of $\PP$  on a flat 
exists under a simple  necessary and sufficient condition
(that is also valid in
symmetric spaces of type $\typeA_{\N-1}$).

%

\begin{prop}
\label{prop- proj p sur A}
Let $\App$ is a flat of $\Es$ 
and $\p\in\PP$. 
Let $(\p_1,\ldots,\p_\N) = (\bordinf\App) \cap \PP$ 
be the points of type $\stypep$ in $\bordinf\App$.
%
There exists a projection of $\p$ on $\App$ 
if and only if
$(\p,\p_1,\ldots,\p_\N)$ is a projective frame.
\end{prop}

The analoguous property is also valid for points $\H\in\PP^*$.

%
\begin{proof}
  If $\p\in H$
  for some hyperplane $\H$ in $\PP^*\cap\bordinf\App$, then
  $\p$ and $\H$ are in a common chamber of the spherical builing
  $\bordinf\Es$, and,
  as the diameter $d$ of the model spherical Weyl chamber $\bord \Cb$ 
  is stricly less that $\pi/2$ (for the angle metric), 
  %
  %
  %
  %
  %
  we have
  $\angleTits(\p,\H)<{\pi/2}$, hence the projection
  do not exist.
  %
  Else, for every hyperplane $\H$ in $\PP^*\cap\bordinf\App$, we have
  $\p\notin H$, hence $\angleTits(\p,\H)=\pi$,
  which implies that   since $\angleTits(\p, \eta) \geq  \pi-d > \pi/2$ 
  for all $\eta\in\bordinf\App$,
  and the projection exist by
  Proposition \ref{prop- CAT0 - projection of ideal  points}.
\end{proof}

%



We now suppose that $\Es$ is a Euclidean building.

\begin{prop}
  \label{prop- center of a projective frame}  
  %
  Let $\Frame=(\p_0, \p_1,\ldots, \p_\N)$  be a projective frame 
  in $\PP\subset\bordinfEs$.
  %
  For each $\indN\in\{0,\ldots, \N\}$ let $\App_\indN$ be the unique 
  flat of $\Es$ through
  $(\p_0,\ldots, \widehat{\p_\indN},\ldots, \p_\N)$.
  %
  There exists a unique point $\xE\in\Es$
  satisfying the following equivalent conditions.

  \begin{enumerate}

  \item 
    \label{i- x in intersection of all flats}
    $\xE \in \cap_\indN \App_\indN$ ;
    %

  \item 
    \label{i-  angles at x are pi}
    %
    For all $\indN$ and for all  $\H$ in $\bordinf\App_\indN\cap\PP^*$  
    %
    the angle $\angle_\xE (\p_\indN,\H)$ is $\pi$ ;
  \item 
    \label{i- directions at x}
    The $(\N+1)$-tuple   $\TangS_\xE\Frame=(\TangS_\xE
    \p_\indN)_{\indN=0,\ldots,\N}$ 
    of directions at $\xE$
    form a projective frame in $\PP_\xE$ ;
    %
    %
    %
    %
    %
    %
    %

  \item 
    \label{i- for all ii, x is the proj on App_ii}
    For all $\indN$, the point  $\xE$ is a (orthogonal) projection 
    of $\p_\indN$ on the flat  $\App_\indN$ ;
    %
    %

  \item 
    \label{i- for one ii, x is the proj on App_ii}
    There exists $\indN$ such that $\xE$ is a 
    projection of $\p_\indN$ on  $\App_\indN$.
    %
  \end{enumerate}


  We will call $\xE$  
  the {\em center}  of the projective frame $\Frame=(\p_0, \p_1,\ldots, \p_\N) $ 
  and denote it by $\centre(\p_0, \p_1,\ldots, \p_\N)$ or $\centre(\Frame)$.
\end{prop}


\begin{figure}[h]
  %
  \includegraphics{figures/Es_ideal_simplex.fig}
\caption{
The center $\xE\in\Es$
of a projective frame  $(\p_1,\p_2,\p_3$, $\p_4)$ (for $\N=3$).
%
}
\end{figure}









\begin{proof}
  The existence of $\xE$ (as a projection of $\p_0$ on  $\App_0$)
  is ensured by  Prop. \ref{prop- proj p sur A}.
  
  For $\indN\neq\indNp$, denote by $H_{\indN\indNp}$ the hyperplane
  $\oplus_{\indNpp\neq\indN,\indNp}\; \p_\indNpp$ in the projective space $\PP$.
  %
  Let $\xE\in\Es$. 
  %
  Conditions (\ref{i- directions at x}) 
  and (\ref{i- x in intersection of all  flats}) 
  are equivalent 
  (see Section \ref{s- x is in App(Frame) iff TangS_xFrame is a frame}).
  %
  %
  %


  We first show the implication
  (\ref{i- x in intersection of all flats})
  $\Rightarrow$ 
  (\ref{i-  angles  at x are pi}):
  %
  Fix $\indN$ and  $\H\in\PP^*$  in $\bordinf\App_\indN$.
  %
  The opposite of $\H$ in $\bordinf\App_\indN$ is some $\p_\indNp$. 
  Then $\H=H_{\indN\indNp}$, 
  so $\H$ is also the opposite of $\p_\indN$
  in the apartment $\bordinf\App_\indNp$. As $\xE\in\App_\indNp$, we then have
  $\angle_\xE (\p_\indN,\H)=\pi$. 
  %
  %
  %
  %
  %
  We now prove 
  (\ref{i-  angles  at x are pi}) 
  $\Rightarrow$
  (\ref{i- directions at x}): 
  %
  First recall that 
  for $\p\in\PP$ and $\H\in\PP^*$, we have 
  $\angle_\xE (\p_\indN,\H)=\pi$ if and only if
  $\TangS_\xE\p\notin\TangS_\xE\H$ in the projective space $\PP_\xE$.
  %
  So (\ref{i-  angles  at x are pi}) means that 
  $\TangS_\xE\p_\indN \notin \TangS_\xE\H_{\indN\indNp}$ for all $\indN\neq\indNp$.
  %
  Let $\U_\indN$ be the minimal linear subspace 
  of the projective space $\PP_\xE$ 
  containing $\TangS_\xE\p_0,\ldots,\TangS_\xE\p_\indN$. 
  Then, for $\indN\leq \N-1$, 
  we have  that $\TangS_\xE\p_\indN$ is not in $\U_{\indN-1}$, 
  else $\TangS_\xE\p_\indN$ would belong to $\TangS_\xE\H_{\indN,\indN+1}$. 
  %
  Hence
  $(\TangS_\xE\p_0,\ldots,\TangS_\xE\p_\indN)$ is generic in $\PP_\xE$ 
  by induction on $\indN$.
  %
  Therefore $(\TangS_\xE\p_0,\ldots,\TangS_\xE\p_{\N-1})$ is a frame, 
  and (\ref{i- directions at x}) follows by permuting the $\p_\indN$.

  %
  We now prove 
  (\ref{i-  angles at x are pi}) 
  $\Rightarrow$
  (\ref{i- for all ii, x is the proj on  App_ii}).
  %
  Let $\indN\in\{0,\ldots,\N\}$.
  Let $\vE\in \TangS_\xE\App_\indN$.
  Let $\Ch\subset\App_\indN$ be a closed Weyl chamber with vertex $\xE$ 
  containing $\vE$. 
  %
  Let $\H\in\PP^*$ be the singular point of type $\stypeH$ in $\bordinf\Ch$. 
  %
  Then $\angle_\xE (\p_\indN,\H)=\pi$, 
  hence $\angle_\xE (\p_\indN, \vE)\geq \pi- d >\frac{\pi}{2}$, 
  as the diameter $d$ of  $\bord \Cb$ 
  is stricly less that $\pi/2$.
  %


  (\ref{i- for all ii, x is the proj on  App_ii})
  $\Rightarrow$ 
  (\ref{i- for one ii, x is the proj on  App_ii})
  is clear.
  %
  Assume now that (\ref{i- for one ii, x is the proj on App_ii}) holds.
  For $\indNp\neq \indN$ in $\{0,\ldots,\N\}$,
  as $\angle_\xE(\p_\indN,\H_{\indN\indNp})\geq \frac{\pi}{2}$,
  the direction $\TangS_\xE\p_\indN$ is not in a
  closed chamber of $\TangS_\xE\Es$ containing $\TangS_\xE\H_{\indN\indNp}$.
  Hence by type considerations we must have $\angle_\xE(\p_\indN,\H_{\indN\indNp})=\pi$.
  So (\ref{i-  angles at x are pi}) holds.

  So the equivalence of all assertions is proven.
  %
  We now prove the uniqueness of $\xE$. 
  Suppose that $\xEp$ is another point of $\Es$ with the same
  properties, and $\xEp \neq \xE$. We proved above that we have then
  $\angle_\xE (\p_\indN, \xEp)> \frac{\pi}{2}$
  and $\angle_{\xEp} (\p_\indN, \xE)> \frac{\pi}{2}$, which is impossible.
  %
\end{proof}


We now state some properties of centers of projective frames.
%
Let $\Frame=(\p_0, \p_1,\ldots,\p_\N)$  
be a projective frame   in $\PP$, 
and let $\xE\in\Es$ be its center.
Let $\App_\indN=\App(\p_0,\ldots, \widehat{\p_\indN},\ldots, \p_\N)$
be the $\N+1$ associated flats in $\Es$.
%
We first describe the intersections of the flats $\App_\indN$ with $\App_0$.

%
\begin{prop}
  \label{prop- Ai cap Aj and partition Ai}
For $\indN=1\ldots\N$, 
let $\SectE_\indN$ be
the \sectorE{} at $\xE$ 
on 
%
$\{\p_1,\ldots, \widehat{\p_\indN},\ldots, \p_\N\}$,
%
i.e.
  the convex hull of the rays from $\xE$ to that points.
%
And let  
$\H_\indN=\p_1 \oplus \cdots \oplus \widehat{\p_\indN} \oplus\cdots \oplus \p_\N$
%
denote the point in $\bordinf\App_0$ opposite to $\p_\indN$.
%
For $\indN \in \{1,\ldots,\N\}$, we have:

\begin{enumerate}

%
\item 
\label{it- dir p_0}
Let $\yE$ be an interior point of $\SectE_\indN$.
%
Then  $\TangS_\yE \p_0 = \TangS_\yE \p_\indN$.
%

\item 
\label{it- p_0 opp H_i}
%
For  $\yE\in\App_0$, we have
$\yE\in\App_0\cap\App_\indN$ 
if and only if    
$\TangS_\yE\p_0$ is opposite to $\TangS_\yE\H_\indN$ ;

%
\item 
%
$\App_0\cap\App_\indN= \SectE_\indN$ ;
\end{enumerate}
%
In particular, the intersections $\App_0\cap\App_\indN$, $\indN=1\ldots\N$,   
form a partition (i.e. a covering with disjoint interiors) of
$\App_\indN$.
%
\end{prop}

\begin{proof}
The  inclusion $\SectE_\indN\subset \App_0\cap \App_\indN$ is clear 
since $\xE\in  \App_0\cap\App_\indN$ 
and $\p_\indNp$ is in 
$\bordinf\App_0  \cap \bordinf \App_\indN$ 
for $\indNp\neq \indN$ in $\{1,\ldots,\N\}$.
%

If  $\yE$ is an interior point of $\SectE_\indN$, then
in the local spherical building $\TangS_\yE\Es$ at $\yE$,
we have that 
$\TangS_\yE\p_0 \in \TangS_\yE\App_0$.
%
Moreover, $\yE\in \App_\indN$ 
as previously observed, 
so $\TangS_\yE\p_0$ is opposite to  $\TangS_\yE\H_\indN$  
(in $\TangS_\yE\App_\indN$).
%
Hence $\TangS_\yE\p_0$ is equal to 
the opposite of $\TangS_\yE\H_\indN$ in $\TangS_\yE\App_0$, which is 
$\TangS_\yE\p_\indN$, proving (\ref{it- dir p_0}).


We now prove (\ref{it- p_0 opp H_i}):
%
In
%
$\PP_\yE$, 
the points 
$(\TangS_\yE\p_1,\ldots, \TangS_\yE\p_\N)$ 
%
form a  frame
(since $\yE\in\App_0$).
%
Hence the $\N-1$ points
%
$(\TangS_\yE\p_1,\ldots,\widehat{\TangS_\yE\p_\indN},\ldots, \TangS_\yE\p_\N)$ 
%
are in generic position.
Therefore  
$(\TangS_\yE\p_0,\ldots,\widehat{\TangS_\yE\p_\indN},\ldots, \TangS_\yE\p_\N)$ 
%
is a frame in $\PP_\yE$
(i.e. $\yE\in\App_\indN$)
if and only if
$\TangS_\yE\p_0 \notin \TangS_\yE\H_\indN$.

%

We finish by proving the remaining inclusion 
$\App_0\cap \App_\indN\subset \SectE_\indN$:
%
The $\SectE_\indN$ clearly form a partition of $\App_0$.
So it is enough to
prove that
 that  $\App_0\cap \App_\indN$ does not meet
the interior of $\SectE_\indNp$ for $\indNp\neq \indN$.
%
Else, at such a point $\yE$,   by (\ref{it- dir p_0}), 
we would have $\TangS_\yE \p_0=\TangS_\yE \p_\indNp$,
which  is not opposite to $\TangS_\yE \H_\indN$, 
providing a contradiction.
\end{proof}


The following Proposition shows that 
the notion of center of projective frames
behaves well with respect to projections to transverse spaces at
infinity.

\begin{prop}
  \label{prop- proj of centers to transverse spaces} 
  %
  %
  %
  For each $\indN$, the projection of $x$ in the transverse building at
  infinity $\Es_{\p_\indN}$ is 
  the center of the projective frame of
  $\bordinf\Es_{\p_\indN}$ formed by the projections
  $\projP_{\p_\indN}(\p_\indNp)=\p_\indN\p_\indNp$ of the $\p_\indNp$, $\indNp\neq \indN$,
  that is:
  %
  %
  $$ \projE_{\p_\indN}(\centre(\p_0, \p_1, \ldots, \p_\N))
  =\centre
  (\p_\i\p_0, \p_\i\p_1,\ldots, \widehat{\p_\i\p_\i},\ldots,\p_\i\p_\N)
  \;.$$
\end{prop}
%

\begin{proof}
  For all  $\indNp\neq \indN$,
  the ray from $\xE$ to $\p_\indN$ 
  is in the flat  $\App_\indNp$ hence
  its projection $\projE_{\p_\indN}(\xE)$ 
  in the transverse building  $\Es_{\p_\indN}$
  is in $\projE_{\p_\indN}(\App_\indNp)$, 
  which is the flat defined by the frame
  $\projP_{\p_\indN}(\p_\indNpp)=\p_\indN\p_\indNpp$, $\indNpp\neq\indN,\indNp$.
  %
  %
  %
\end{proof}




%
In the algebraic case, 
i.e. when  $\Es$ is the Euclidean buiding $\Es(\V)$ 
associated with some vector space $\V$ of dimension $\N$
 over an ultrametric field $\KK$, 
%
we have the following characterisation of the center 
%
as a norm on $\V$.
%

\begin{prop}
  Let $\Frame=(\p_0, \p_1,\ldots, \p_\N)$  be a projective frame in $\PP=\PV$.
  The center  of $\Frame$
  %
  is the norm $\normV$ on $\V$ canonically associated 
  to any basis $\eVb=(\eV_\indN)_{\indN=1,\ldots \N}$ of $\V$ such that
  $\p_\indN=\classPV{\eV_\indN}$ for $1\leq \indN \leq \N$ and
  $\p_{0}=\classPV{\eV_1+\cdots+\eV_\N}$ in $\PV$, 
  i.e. the norm defined by 
  $$\normV(\sum_{\indN=1}^\N\aK_\indN\eV_\indN)=\max_{1\leq \indN \leq \N}\abs{\aK_\indN}\;.$$
\end{prop}


\begin{proof}
  %
  %
  %
  Let $\eVb=(\eV_1,\ldots,\eV_\N)$ be a basis of $\V$ such that
  $\p_\indN=\classPV{\eV_\indN}$ and
  $\p_{0}=\classPV{\eV_1+\cdots+\eV_\N}$ in $\PV$.
  %
  Let $\normV$ 
  be the associated canonical norm on $\V$.
  %
  %
  %
  We clearly have $\normV \in \App_{0}$ by the definition of marked
  flats in the model of norms.
  Let $\g$ be the element of
  $\GL(\V)$ sending the basis $\eVb$ to the basis
  $(\eV_1,\ldots,\eV_{\N-1},\eV_1+\cdots+\eV_\N)$.
  Then $\g$ 
  preserves the norm $\normV$ 
  and sends $\App_0$ to $\App_\N$ and 
  %
  hence $\normV$ is in the flat $\App_\N$.
  %
  Permuting the basis $\eV$, we similarly get  that $\normV$ is in 
  the flat $\App_\indN$ for all $\indN\neq 0$.
  %
  %
\end{proof}









\begin{rema}
  By duality, the similar properties hold for generic $(\N+1)$-tuples
  (projective frames)
  in  $\PP^*\subset\bordinf\Es$.
\end{rema}





%

\subsection{Projecting  two ideal points on a flat}
%
From now on we return to the case where $\N=3$ (type $\typeA_2$).
%

%

%
\begin{prop}
\label{prop- 1 plat, 2 points}
Let $(\p_1,\p_2,\p_3)$ be a generic triple in $\PP$.
%
Let $\p,\q$ be two points in $\PP$, 
in generic position relatively to the $\p_\i$ 
(i.e. not on any of the lines $\p_\i\p_\j$).
%
Denote by $\xE$ and $\yE$ the respective projections of $\p$ and $\q$ on
the flat $\App=\App(\p_1,\p_2,\p_3)$.
%
Identify $\App$ with $\Aa$ by a marked flat
sending $\bord\Cc$ to $(\p_1,\p_1\p_2)$.
%
Then the roots coordinates of $\ovra{\xE \yE}$ are given by the three
natural cross ratios at the vertices of the triangle:
%
$$\rac1( \ovra{\xE \yE})
%
=  \geombir(\p_3\p_1, \p_3\p, \p_3\p_2 , \p_3\q),$$
%
$$\rac2( \ovra{\xE \yE})
= \geombir(\p_1\p_2, \p_1\p,  \p_1\p_3 , \p_1\q ),$$
%
$$\rac3( \ovra{\xE \yE})
= \geombir(\p_2\p_3, \p_2\p,  \p_2\p_1 , \p_2\q )\;.$$
\end{prop}

%

\begin{proof}
Projecting on the transverse tree $\Es_{\p_1}$ in direction $\p_1$,
 we have
$$\rac2( \ovra{\xE \yE})
=\rac2( \Bus{(\p_1,\p_1\p_2)} {\xE} {\yE} )
=\Bus{\p_1\p_2} {\projE_{\p_1}(\xE)} {\projE_{\p_1}(\yE)}$$
by  
\eqref{eq- projections and Busemann cocycle}.
%
Since the projections of $\xE$ and $\yE$ on the tree $\Es_{\p_1}$ are the
respective centers of the ideal triples $(\p_1\p_2, \p_1\p_3, \p_1\p)$
and $(\p_1\p_2, \p_1\p_3, \p_1\q)$ 
(Proposition \ref{prop- proj of centers to transverse spaces}),
%
we have 
$$\Bus{\p_1\p_2} {\projE_{\p_1}(\xE)} {\projE_{\p_1}(\yE)}
=  \geombir(\p_1\p_2, \p_1\p,  \p_1\p_3 , \p_1\q )$$
by \eqref{eq- geombir and centers of tripods},
hence
$\rac2( \ovra{\xE \yE})
= \geombir(\p_1\p_2, \p_1\p,  \p_1\p_3 , \p_1\q )$.
%
The remaining assertions follow by applying cyclic permutation, since 
$$\rac1(\Bus{(\p_1,\p_1\p_2)}{\xE}{\yE})
=\rac2(\Bus{(\p_3,\p_3\p_1)}{\xE}{\yE})$$
$$\rac3(\Bus{(\p_1,\p_1\p_2)}{\xE}{\yE})
=\rac2(\Bus{(\p_2,\p_2\p_3)}{\xE}{\yE})\; .$$
%
\end{proof}

The analogous dual result holds for projections of two lines (exchanging
the roles of points and lines in $\PP$).
%
And for the projections of a point and a line, we have the following result.

%

\begin{prop}
\label{prop- 1 flat p D}
  Let $\F_-=(\p_-,\D_-)$ et $\F_+=(\p_+,\D_+)$ be two oppposite flags
  in $\PP$
and $\App$ the flat in $\Es$ joining them, identified with $\Aa$
by a marked flat sending $\bord\Cc$ to $\F_+$.
%
Let $\p$ be a point and $\D$ a line 
in $\PP$ in generic position with respect to $\F_-$ and $\F_+$.
%
Denote by $\xE$ and $\xE^*$ the respective projections of $\p$ and $\D$ 
on $\App$. Then in simple roots coordinates we have 
$$\ovra{\xE \xE^*}=(\gZ_-,\gZ_+),$$
\begin{align*}
\mbox{with }
\gZ_-
%
&=  \geombir(\p_+, \D_+\cap (\p_-\p),  \D_+\cap\D_- , \D_+\cap \D)\\
%
&=  \geombir(\D_-, \p_- \oplus (\D_+\cap \D),  \p_-\p_+ , \p_-\p )\\
%
\mbox{and }
\gZ_+
%
&=  \geombir(\p_-, \D_-\cap \D,  \D_-\cap\D_+ , \D_-\cap (\p_+\p))\\
%
&=  \geombir(\D_+, \p_+\p,  \p_+\p_- , \p_+ \oplus (\D_-\cap \D) )
\;.\end{align*}
%
\qed
\end{prop}

%

\begin{proof}
The projection of $\xE$ on the transverse tree $\Es_{\p_-}$ 
is  the center of the ideal triple 
$(\p_-\p_+, \p_-(\D_-\cap\D_+), \p_-\p)$, and
the projection of $\xE^*$ on the tree $\Es_{\D_+}$ 
is  the center of the ideal triple 
$(\p_+, \D_+\cap\D_-,\D_+\cap\D)$ 
(Proposition \ref{prop- proj of centers to transverse spaces}).
%
As $\xE$ lies on a geodesic from $\p_-$ to $\D_+$, 
we have
\begin{align*}
\projE_{\D_+}(\xE) 
&= \projE_{\D_+,\p_-} ( \projE_{\p_-}(\xE) )\\
&= \projE_{\D_+,\p_-} ( \centre(\p_-\p_+, \p_-(\D_-\cap\D_+), \p_-\p) )\\
&=\centre (\p_+, \D_-\cap\D_+, \D_+ \cap (\p_-\p) )
\end{align*}
by \eqref{eq- projE_Dp preserves centers}.
%
Then projecting on the transverse tree $\Es_{\D_+}$ we have
$$\rac1( \ovra{\xE \xE^*})
= \Bus{\p_+}{\projE_{\D_+}(\xE)} {\projE_{\D_+}(\xE^*)}
=  \geombir(\p_+, \D_+\cap (\p_-\p),  \D_+\cap\D_- , \D_+\cap \D)$$
as needed. The remaining assertions have identical proofs.
\end{proof}


%

%

%



%

triples_3_triples_of_ideal_chambers.tex
%



%

\section{Triple ratio of a triple of ideal chambers%
%
}
\label{s- triple ratio}
%

%
In this section, we introduce  
the {\em (geometric) triple ratio} of a \nondegeneratedTF{} 
triple of ideal chambers in a
a real Euclidean building $\Es$ of type $\typeA_2$,
%
establish its basic properties, 
%
and the links with the
usual $\KK$-valued (algebraic) triple ratio of triples of flags
(see e.g. \cite{FoGoSPC})
in the algebraic case $\PP=\PP(\KK^3)$.


We first precise 
the notions of {\em \nondegeneratedTF{}} and {\em \genericTF} triples of flags 
in an arbitrary projective plane  $\PP$.

%
\subsection{\NondegeneratedTF{} triples of flags}
\label{s- nondegeneratedTF}
%
Let $\PP$ be a projective plane
and $\Tau=(\F_1,\F_2,\F_3)$ be a triple of flags
 $\F_\ii=(\p_\ii,\D_\ii)$ in $\PP$.
%
We will  denote by $\p_\ij$ the point $\D_\i\cap \D_\j$
 (resp. $\D_\ij$ the line $\p_\i\p_\j$), when defined.

%
The natural nondegeneracy condition on the triple $(\F_1,\F_2,\F_3)$ 
for the triple ratios  to be well defined is the following:
%
\begin{center}
(\DefTR)
either for all $\ii$, $\p_\ii \notin \D_{\ii+1}$ 
or for all $\ii$, $\p_\ii \notin \D_{\ii-1}$. 
\end{center}
%
This condition is clearly equivalent to:
the points are pairwise distinct, the lines are pairwise distinct,
none of the points is on the three lines 
(i.e. $\D_\ii\cap \D_\jj \neq \p_\kk$ for all $\{\ii,\jj,\kk\}=\{1,2,3\}$)
and none of the lines contains the three points
(i.e. $\p_\ii \p_\jj \neq \D_\kk$ for all $\ii,\jj,\kk$). 
%
We will then say that the triple $(\F_1,\F_2,\F_3)$ is 
{\em \nondegeneratedTF}.

%
It is easy to check that the triple $\Tau$ defines then 
a \nondegeneratedQ{} quadruple of well-defined lines
$\D_\ii$, $\p_\ii\p_\jj$, $\p_\ii\p_\jk$, $\p_\ii\p_\kk$ 
through each point $\p_\ii$,
and a \nondegeneratedQ{} quadruple of well-defined points 
$\p_\ii$, $\D_\ii\cap\D_\jj$, $\D_\ii\cap\D_\jk$, $\D_\ii\cap\D_\kk$ 
on each line $\D_\ii$.
%



%
The  triple of flags  $\Tau=(\F_1,\F_2,\F_3)$ is {\em \genericTF} if
the flags $\F_\ii=(\p_\ii,\D_\ii)$ are pairwise opposite, 
the points $(\p_\ii)_\ii$ are not collinear 
and the lines  $(\D_\ii)_\ii$ are not concurrent.
In particular, $\Tau$ is then \nondegeneratedTF{}, and the induced
quadruples of points on each line (resp. of lines through each point)
are generic (pairwise distinct).

%

\subsection{Algebraic triple ratio}

%
When $\PP=\PP(\KK^3)$ is the projective plane associated 
with an arbitrary field $\KK$,
 the algebraic triple ratio of a \nondegeneratedTF{} 
triple of flags $\Tau=(\F_1,\F_2,\F_3)$ 
(see section \ref{s- nondegeneratedTF}), 
with values in $\KK\cup\{\infty\}$,  is defined by
(see  \cite[\S 9.4  p128]{FoGoIHES})
%
$$
\Tri(\F_1,\F_2,\F_3)
=
\frac{\Dt_1(\pt_2)\Dt_2(\pt_3)\Dt_3(\pt_1)}{\Dt_1(\pt_3)\Dt_2(\pt_1)\Dt_3(\pt_2)}
$$
%
where $\pt_\ii$ is any vector in $\KK^3$ representing $\p_\ii$ and
$\Dt_\ii$ is any linear form in $(\KK^3)^*$ representing $\D_\ii$, and
$\F_\ii=(\p_\ii,\D_\ii)$.
%
It is invariant under cyclic permutation of the flags 
and inversed by reversing the order
\[\Tri( \F_3, \F_2, \F_1 )=\Tri(\F_1,\F_2,\F_3)^{-1} \;.\]
%
It may be expressed as the following cross ratio
%
\begin{align}
\label{eq- triple ratio as a cross ratio}
\Tri(\F_1,\F_2,\F_3)
&=\Bir(\D_1, \p_1 \p_2, \p_1 \p_{23}, \p_1 \p_3)\;.
%
\end{align}
%



\subsection{Geometric triple ratio}
%

We suppose now that the projective plane $\PP$ is the projective plane
at infinity of some a real Euclidean building $\Es$ of type
$\typeA_2$. Let $\geombir$ be the associated geometric cross ratio on $\PP$ 
(see section \ref{ss- geometric cross ratio on a projective plane}).
%
Let   $\Tau=(\F_1,\F_2,\F_3)$ be a \nondegeneratedTF{} 
triple of ideal chambers of $\Es$,
i.e. a \nondegeneratedTF{}  triple of flags $\F_\ii=(\p_\ii,\D_\ii)$ in $\PP$. 

%
We define the {\em geometric triple ratio} 
of  $\Tau$,
by analogy of the algebraic triple ratio expressed as a cross ratio
\eqref{eq- triple ratio as a cross ratio},
as the following triple of geometric cross ratios in $\PP$,
obtained from the induced quadruple of lines $\D_1, \p_1\p_2,
\p_1\p_{23}, \p_1\p_3$ at $\p_1$
by cyclic permutation of the three last one. 
%
\[
\begin{array}{rcl}
  \geomtrio(\F_1,\F_2,\F_3) &=& 
  \geombir(\D_1, \p_1\p_2, \p_1\p_{23}, \p_1\p_3)\\
%
  \geomtrip(\F_1,\F_2,\F_3)&=&
%
  \geombir(\D_1, \p_1\p_3, \p_1\p_2, \p_1\p_{23})\\
%
  \geomtripp(\F_1,\F_2,\F_3)&=& 
%
  \geombir(\D_1, \p_1\p_{23}, \p_1\p_3, \p_1\p_2)
\end{array}
\]
%
Note these cross ratios are well defined, since the four lines
$\D_1$, $\p_1\p_2$, $\p_1\p_{23}$, $\p_1\p_3$ 
are well defined and form a \nondegeneratedQ{} 
quadruple (see section \ref{s- nondegeneratedTF} above).
%
The {\em geometric triple ratio} of $\Tau$ 
is $\geomtri(\Tau)=(\geomtri_\indTriRap(\Tau))_{\indTriRap=1,2,3}$ in $\RR^3$.


%
The following proposition gathers the properties of the geometric
triple ratio.
  
\begin{prop}
  \label{prop- props of geomtri}
The following hold.
  \begin{enumerate}
    %
  \item 
    \label{item- geomtri: invariant by cyclic perm}
    The geometric triple ratio is invariant by cyclic permutations of
    the flags, i.e.
    for $\indTriRap=1,2,3$, 
    $$\geomtri_\indTriRap(\F_2,\F_3,\F_1)
    =\geomtri_\indTriRap(\F_1,\F_2,\F_3) \; ;$$ 


  \item Exchanging two flags, we have
    \label{item- geomtri: exchanging two flags}
     %
     %
     %
 %
\[
    \begin{array}{rl}
      \geomtrio(\F_1,\F_3,\F_2)&=-\geomtrio(\F_1,\F_2,\F_3),\\  
      \geomtrip(\F_1,\F_3,\F_2)&=-\geomtripp(\F_1,\F_2,\F_3)
    \end{array}
    \; ;\]

%

  \item
    \label{item- sum geomtri is 0}
 We have $\geomtrio(\Tau)+\geomtrip(\Tau)+\geomtripp(\Tau)=0$; 


    %
  \item  
    \label{item- geomtri is ultram} 
%
    for all $\indTriRap\in\ZZ/3\ZZ$,
    if $\geomtri_\indTriRap(\Tau)>0$,
    then $\geomtri_{\indTriRap-1}(\Tau)=0$ and 
    $\geomtri_{\indTriRap+1}(\Tau)=-\geomtri_\indTriRap(\Tau)<0$.

  \end{enumerate}
\end{prop}

%


\begin{proof}


Assertions (\ref{item- sum geomtri is 0}) 
and (\ref {item- geomtri is ultram})
follow immediately 
from the properties of the cross ratio $\geombir$ 
under cyclic permutation of the three last points
(see \eqref{eq- sum of 3-cyclic perm of geom bir is 0} 
and
%
 \eqref{eq- geombir is ultram}).

%


Assertion (\ref{item- geomtri: exchanging two flags}) 
follows immediately from the definition and from the symmetries of
the cross ratio.


%
In order to prove the invariance of the triple ratio by cyclic
permutation of the flags 
(\ref{item- geomtri: invariant by cyclic  perm}),
%
a nice way is to introduce 
the natural dual invariants given by the cross ratios 
of the natural induced quadruple of points on the line $\D_1$ 
(that is, exchanging the role of points and lines):
%
$$    \begin{array}{rl}
\geomtrieto(\F_1,\F_2, \F_3) 
&=  \geombir(\p_1,\D_2 \cap \D_1, \D_{23} \cap \D_1, \D_3 \cap \D_1)\\
%
\geomtrietp(\F_1,\F_2,\F_3)
&=  \geombir(\p_1, \D_3\cap\D_1, \D_2 \cap \D_1, \D_{23} \cap \D_1)\\
%
\geomtrietpp(\F_1,\F_2,\F_3)
&= \geombir(\p_1, \D_{23} \cap \D_1, \D_3\cap\D_1, \D_2 \cap \D_1)\;.
    \end{array}
$$
%

%

The following property is straigthforward.
%
\begin{equation}
  \label{eq- geomtriet: exchanging two flags}
  \begin{array}{rl}
    \geomtrieto(\F_1,\F_3,\F_2)&=-\geomtrieto(\F_1,\F_2,\F_3),\\  
    \geomtrietp(\F_1,\F_3,\F_2)&=-\geomtrietpp(\F_1,\F_2,\F_3)
%
  \end{array}
\end{equation}


%

We now show that the invariants behave nicely under duality.

\begin{lemm}
\label{lemm- geomtriet and geomtri}
 
For $\indTriRap=1,2,3$, we have 
$\geomtriet_\indTriRap(\F_1,\F_2, \F_3)
=  \geomtri_\indTriRap(\F_3,\F_2,\F_1)$.
  
\end{lemm}

\begin{proof}[Proof of Lemma \ref{lemm- geomtriet and geomtri}]
  %
By invariance under perspectivities and double transpositions,
we have
%
\begin{align*}
\geomtrieto(\F_1,\F_2,\F_3) 
&=  \geombir(\p_1, \D_2\cap\D_1, \D_{23}\cap\D_1, \D_3\cap\D_1)\\
&=  \geombir(\p_1\p_3, \p_{12}\p_3, \D_{23}, \D_3)\\
%
&=  \geombir(\D_3, \p_2\p_3, \p_{12}\p_3, \p_1\p_3)\\
%
&= \geomtrio(\F_3,\F_2, \F_1)\;.
\end{align*}
%
Applying cyclic permutation of the
last three  arguments to both sides of the equality 
$$ \geombir(\p_1, \D_2\cap\D_1, \D_{23}\cap\D_1, \D_3\cap\D_1) 
= \geombir(\D_3, \p_2\p_3, \p_{12}\p_3, \p_1\p_3),$$
we obtain that
$\geomtriet_\indTriRap(\F_1,\F_2, \F_3)= \geomtri_\indTriRap(\F_3,\F_2,\F_1)$ for $\indTriRap=2,3$.
\end{proof}

 
We now finally 
prove Assertion (\ref{item- geomtri: invariant by cyclic  perm})
of Proposition \ref{prop- props of geomtri}.
%
Using 
Assertion (\ref{item- geomtri: exchanging two flags}), 
Lemma \ref{lemm- geomtriet and geomtri}
and
\eqref{eq- geomtriet: exchanging two flags},
we have
  \begin{align*}
    \geomtrio(\F_2,\F_3,\F_1)
    &=-\geomtrio(\F_2,\F_1,\F_3)\\ 
    &=-\geomtrieto(\F_3,\F_1, \F_2)\\
    &=\geomtrieto(\F_3,\F_2, \F_1)
    =\geomtrio(\F_1,\F_2,\F_3) ,
  \end{align*}
  \begin{align*}
    \geomtrip(\F_2,\F_3,\F_1)
    &=-\geomtripp(\F_2,\F_1,\F_3)\\ 
    &=-\geomtrietpp(\F_3,\F_1, \F_2)\\
    &=\geomtrietp(\F_3,\F_2, \F_1)
    =\geomtrip(\F_1,\F_2,\F_3)\;.
  \end{align*}
The case where $\indTriRap=3$ is similar to the case $\indTriRap=2$.
\end{proof}



















\subsection{Geometric triple ratio from  algebraic triple ratio}
%
When $\PP$ is the projective plane  
on some field $\KK$  endowed with some ultrametric absolute value, 
 and $\geombir = \log\abs{\Bir}$ where $\Bir$ 
is the usual $\KK$-valued cross ratio on $\PP$,
the three geometric triple ratios $\geomtri_\indTriRap(\Tau)$, $\indTriRap=1,2,3$
of $\Tau$
 are obtained from 
the single algebraic triple ratio $\Z=\Tri(\Tau)$ of $\Tau$
 by  the following relations 
%
\begin{equation}
  \begin{array}{rcl}
    \geomtrio(\Tau)&=&\log\abs{\Z}\\
    \geomtrip(\Tau) &=&\log\abs{\frac{1}{1+\Z}}=-\log\abs{1+\Z}\\
    \geomtripp(\Tau) &=& \log\abs{1+\Z^{-1}},
  \end{array}
\end{equation}
%
which are easily derived from the expression of 
algebraic triple ratio as a  cross ratio \eqref{eq- triple ratio as a cross ratio}
and
from the symmetry properties of the algebraic cross ratio  \eqref{eq- sym bir 1342}.
%





\begin{rema}
Note that the geometric invariants do not determine 
the triple of flags up to automorphisms of $\PP$
%
(unlike the usual (algebraic) triple ratio): 
%
for example in the algebraic case  $\PP=\PP(\KK^3)$,  
take $\Tau$ with triple ratio $\Z\in\KK$ with $\abs{\Z}>1$ 
and $\Taup$ with triple ratio $\Zp=\Z\aK$ where $\aK\in\KK$ with $\abs{\aK}=1$ and
$\aK\neq 1$. Then $\Tau$ and $\Taup$ are not in the same
$\PGL(\KK^3)$-orbit, but have the same three geometric
invariants, as
$\geomtrio(\Tau) = \log\abs{\Z}=\geomtrio(\Taup)$, 
$\geomtrip(\Tau) =-\log\abs{\Z}=\geomtrip(\Taup)$,
$\geomtripp(\Tau)= 0=\geomtripp(\Taup)$.
%
\end{rema}




























\section{Proof of the main result}
In this section, we study the geometry in an $\typeA_2$-Euclidean
building $\Es$ of a generic triple of ideal chambers, and
prove Theorems \ref{theo- tripod} and \ref{theo- triangle}.
%
From now on,
we suppose that 
$\Tau=(\F_1,\F_2,\F_3)$  is a \genericTF{} triple of flags 
in the projective plane $\PP$ at infinity of $\Es$.
%
We denote by 
$\gZ_\indTriRap=\geomtri_\indTriRap(\F_1,\F_2,\F_3)$, 
$\indTriRap=1,2,3$,
its geometric triple ratio.
%
Recall that $\Aij=\App(\F_\ii,\F_\jj)$, $\Ap=\App(\p_1,\p_2,\p_3)$ and
$\AD=\App(\D_1,\D_2,\D_3)$ denote the five associated flats.



%



 \subsection{Associated  points in the building}
%
For $\{\ii,\jj,\kk\}=\{1,2,3\}$,
%
denote by $\yTauE_\kk$  the center in $\Es$ of the projective frame
$\p_1,\p_2,\p_3, \p_\iijj=\D_\ii \cap \D_\jj$,
%
and by $\yTauE_\kk^*$   the center of the projective frame
$\D_1,\D_2,\D_3, \D_\iijj=\p_\ii\p_\jj$,
%
as defined in 
Proposition \ref{prop- center of a projective frame}.
%
In particular
the point $\yTauE_\kk$ is the (orthogonal) projection of $\p_\iijj$ on $\Ap$,
the point $\yTauE_\kk^*$ is the projection of $\D_\iijj$ on  $\AD$,
the point $\yTauE_\kk$ is the  projection of $\p_\kk$ 
on $\Aij=\App(\p_\ii,\p_\jj,\p_\iijj)$, 
and
the point $\yTauE_\kk^*$ is the projection of $\D_\kk$ 
on $\Aij=\App(\D_\ii,\D_\jj,\D_\iijj)$.
%

%


\subsection{In the flat \texorpdfstring{$\Aij$}{A\_ij}}
Here we link 
the respective position of $\yTauE_\kk$ and $\yTauE_\kk^*$ in the flat
$\Aij$ to the geometric triple ratio of $\Tau$. 
%
Suppose that the indices $\ii,\jj,\kk$ respects the cyclic order, 
i.e. that $(\ii,\jj,\kk)=(1 2 3)$ as cyclic permutations.
%
We identify $\Aij$ with the model flat $\Aa$ 
by a marked flat $\f_\iijj:\Aa \to \Aij$ sending $\bord\Cc$ to
$\F_\jj$.
%
For $\xE,\yE$ in $\Aij\simeq\Aa$, 
we define then $\ovra{\xE\yE}=\yE-\xE=\Bus{\F_\jj}{\xE}{\yE}$.
%
Recall that 
$(\eRRNo_1,\eRRNo_2,\eRRNo_3)$ denotes the canonical basis of $\RR^3$.
In particular, the  directions of $\p_\ii$, $\p_\ij$ and $\p_\jj$  
are respectively identified with (the directions of) 
$\classA{\eRRNo_1}$, $\classA{\eRRNo_2}$, and 
direction $\classA{\eRRNo_3}$.

\begin{prop}
\label{prop- in Aij - y*y}
The following holds.
%
\begin{enumerate}
\item 
\label{i- y*y in simple roots coords} 
in simple roots coordinates, we have
$\ovra{\yTauE_\kk^* \yTauE_\kk}= (\gZp, \gZpp)$;

%

\item 
\label{i- yy* sing of type point} 
%
for $\indTriRap=1,2,3$, if $\gZ_\indTriRap>0$
%
then 
$\ovra{\yTauE_\kk\yTauE_\kk^*}=\gZ_\indTriRap
\classA{\eRRNo_\indTriRap}$. 
%
In particular
$\yTauE_\kk^*$ is on one of the three singular rays of type
  $\stypep$ issued from $\yTauE_\kk$ 
(i.e the rays to $\p_\ii$,  $\p_\jj$ and $\p_\ij$).
%
\end{enumerate}
\end{prop}



\begin{proof}
%
As $\yTauE_\kk$ and $\yTauE_\kk^*$ are the respective projections on the flat
$\App_\ij$ of $\p_\k$ and $\D_\k$, 
by Proposition \ref{prop- 1 flat p D} and  cyclic invariance of the geometric triple ratio, we have
\begin{align*}
&
\rac1( \ovra{\yTauE_\kk^* \yTauE_\kk})
=  \geombir(\D_\i, \p_\i\p_\k,  \p_\i\p_\j , \p_\i \p_\jk )
= \geomtri_2(\F_\i, \F_\j, \F_\k)
= \gZp\\
\mbox{and }
&
\rac2( \ovra{\yTauE_\kk^* \yTauE_\kk})
=  \geombir(\D_\j, \p_\j \p_\ki,  \p_\j\p_\i , \p_\j\p_\k )
= \geomtri_3(\F_\j, \F_\k, \F_\i)=\gZpp
\;.
\end{align*}

%


%
Assertion (\ref{i- yy* sing of type point}) 
follows, since we have then 
$\gZ_{\indTriRap-1}=0$ and $\gZ_{\indTriRap+1}=-\gZ_\indTriRap$ 
by ultrametricity of the geometric triple ratio
(Proposition \ref{prop- props of geomtri}(\ref{item- geomtri is
  ultram})).
%
\end{proof}


We now describe the intersections of $\Aij$ with the four other flats
(see figures \ref{fig-  type tripod} and 
\ref{fig-  type triangle in positive case} in the introduction). 

%

\begin{prop}
\label{prop- in Aij - intersections}
%
Let $\xE\in\Aij$. Then

\begin{enumerate}

\item
\label{i- Aij cap Ap}
The intersection $\Aij\cap\Ap$ is 
the \sectorE{} 
%
at $\yTauE_\kk$  
bounded by the rays to $\p_\ii$ and $\p_\jj$.
That is
$$\xE\in  \Ap
\mbox{  if and only if }
\left\{
\begin{array}{lll}
\rac1(\xE) & \geq & \rac1(\yTauE_\kk) \\
\rac2(\xE) & \leq & \rac2(\yTauE_\kk)
\end{array}
\right.
\;.$$

%

%



\item
\label{i- Aij cap AD}
The intersection $\Aij\cap\AD$ is 
the \sectorE{} at $\yTauE_\kk^*$  
bounded by the rays to $\D_\ii$ and $\D_\jj$.
That is, 
$$\xE\in  \AD
\mbox{  if and only if }
\left\{
\begin{array}{lll}
\rac1(\xE) & \leq & \rac1(\yTauE_\kk^*) \\
\rac2(\xE) & \geq & \rac2(\yTauE_\kk^*)
\end{array}
\right.
\;.$$
%

\item
\label{i- Aij cap Ajk}
The intersection $\Aij\cap\Ajk$ is the intersection of 
%
the \sectorE{} at $\yTauE_\kk$ bounded by the rays to $\p_\jj$ and
$\D_\ii\cap\D_\jj$,
and
%
the \sectorE{} at $\yTauE_\kk^*$ bounded by the rays to $\D_\jj$ and
$\p_\ii\p_\jj$. 
%
That is,
%
$$\xE\in  \Ajk
\mbox{  if and only if }
\left\{
\begin{array}{lll}
\rac1(\xE) & \geq &  \rac1(\yTauE_\kk^*) \\
\rac2(\xE) & \geq & \rac2(\yTauE_\kk) \\
\rac3(\xE) & \leq & \min(\rac3(\yTauE_\kk), \rac3(\yTauE_\kk^*))
\end{array}
\right.
\;.$$


\item
\label{i- Aij cap Aki}
The intersection $\Aij\cap\Aki$ is the intersection of 
%
the \sectorE{} at $\yTauE_\kk$ bounded by the rays to $\p_\ii$ and
$\D_\ii\cap\D_\jj$,
and
%
the \sectorE{} at $\yTauE_\kk^*$ bounded by the rays to $\D_\ii$ and
$\p_\ii\p_\jj$.
%
That is,
$$\xE\in  \Aki
\mbox{  if and only if }
\left\{
\begin{array}{lll}
\rac1(\xE) & \leq &  \rac1(\yTauE_\kk) \\
\rac2(\xE) & \leq & \rac2(\yTauE_\kk^*) \\
\rac3(\xE) & \geq & \max(\rac3(\yTauE_\kk), \rac3(\yTauE_\kk^*))
\end{array}
\right.
\;.$$
%
\end{enumerate}
\end{prop}




\begin{proof}
Since $\yTauE_\kk$ is the center of the projective frame
$(\p_\ii,\p_\jj, \p_\ij,\p_\kk)$, 
Assertion (\ref{i- Aij cap Ap}) comes from
Proposition \ref{prop- Ai cap Aj and partition Ai}, as
$\App_\ij=\App(\p_\ii,\p_\jj, \p_\ij)$
and  $\Ap=\App(\p_\ii,\p_\jj,\p_\kk)$.
%
Assertion (\ref{i- Aij cap AD}) is similar. 
%
Assertion (\ref{i- Aij cap Ajk}):
A point $\xE\in\Aij$ lies in $\Ajk$ 
if and only if, 
in the spherical building of directions at $\TangS_\xE\Es$, 
the direction $\TangS_\xE\D_\jj$ is  opposite to $\TangS_\xE\p_\kk$  
and  $\TangS_\xE\p_\jj$ is  opposite to $\TangS_\xE\D_\kk$.
%
Moreover, 
$\TangS_\xE\D_\jj$ is  opposite to $\TangS_\xE\p_\kk$
%
if and only if
$\xE\in\App(\p_\kk,\p_\jj,\p_\iijj)$.
%
As $\yTauE_\kk$ is the center of the projective frame 
$(\p_\ii,\p_\jj,\p_\iijj, \p_\kk)$ and 
$\App_\ij=\App(\p_\ii,\p_\jj,\p_\iijj)$,
the set of such $\xE$ is
the \sectorE{} with vertex $\yTauE_\kk$ bounded by 
the rays to $\p_\jj$ and $\D_\ii\cap\D_\jj$
(by Proposition \ref{prop- Ai cap Aj and partition Ai}).
%
This is the subset of $\xE\in\Aij$ satisfying:
$\rac2(\xE)\geq \rac2(\yTauE_\kk)$ and
$\rac3(\xE)\leq\rac3(\yTauE_\kk)$.
%
Similarly 
As $\yTauE^*_\kk$ is the center of the projective frame 
$(\D_\ii,\D_\jj,\D_\iijj, \D_\kk)$ and 
$\App_\ij=\App(\D_\ii,\D_\jj,\D_\iijj)$,
the direction $\TangS_\xE\p_\jj$ is  opposite to $\TangS_\xE\D_\kk$
%
if and only if
$\xE$ is in the \sectorE{} 
with vertex $\yTauE_\kk^*$ 
bounded by the rays to $\D_\jj$ and $\D_\ij=\p_\ii\p_\jj$.
%
That is, if and only if
$\rac1(\xE)\geq \rac1(\yTauE_\kk^*)$ and
$\rac3(\xE)\leq\rac3(\yTauE_\kk^*)$, and we are done.
%
Assertion (\ref{i- Aij cap Aki}) is similar.
\end{proof}


In particular, as $\yTauE_\kk^*$ is on one of the three singular rays of type
  $\stypep$ issued from $\yTauE_\kk$ by Propositions \ref{prop- in Aij - y*y}, 
from Proposition \ref{prop-  in Aij - intersections} we easily 
%
get the following result.


\begin{coro}
\label{coro- partition Aij}
The intersections with $\Aij$ of $\Ajk$,$\Aki$, $\Ap$ and $\AD$ 
form a partition  of $\Aij$.  
\qed\end{coro}
%


\subsection{In the flat \texorpdfstring{$\Ap$}{A\_p}}
%
We now consider the flat $\Ap=\App(\p_1,\p_2,\p_3)$.
%
The following Proposition describes the respective positions in $\Ap$
of the points $\yTauE_1$, $\yTauE_2$, $\yTauE_3$.
%
We identify $\Ap$ with $\Aa$ by a marked flat $\f_\p:\Aa \to
\Ap$ sending $\bord\Cc$ to $(\p_1,\p_1\p_2)$
(hence direction $\classA{\eRRNo_\ii}$ to $\p_\ii$ for $\ii=1,2,3$). 
%
Recall that we then have
$\ovra{\xE\xE'}=\xE'-\xE=\Bus{(\p_1,\p_1\p_2)} {\xE}{\xE'}$  for
$\xE,\xE'\in\Ap$.


\begin{prop}
\label{prop- in Ap}
%
In the flat $\Ap$ we have:


 \begin{enumerate}
\item
 \label{i- y2 y3}
In simple roots coordinates, we have $\ovra{\yTauE_2\yTauE_3}=(\gZo,0)$.
%

 \item 
\label{it- y_i y_i+1}
if $\gZo\geq 0$,
the point  $\yTauE_{\i+1}$ is in the ray $[\yTauE_\ii,\p_{\ii+2})$ 
(for all $\ii$),
and if $\gZo\leq 0$,
the point $\yTauE_\ii$ is in the ray $[\yTauE_{\ii+1},\p_{\ii+2})$ for
all $\ii$.

%

%




%
\end{enumerate}

In particular the triangle $\Delta\subset \Ap$ 
with vertices $\yTauE_1$, $\yTauE_2$,  $\yTauE_3$ 
is {\em  singular}, 
i.e.the sides have singular type in $\Cb$.
\end{prop}


\begin{proof}
%
Recall that the point $\yTauE_\kk$ is the orthogonal projection 
on the flat $\Ap$ 
of the singular boundary point $\p_\iijj=\D_\ii \cap \D_\jj$.
%
Then, by Proposition \ref{prop- proj of centers to transverse spaces}
the points  $\yTauE_2$ and $\yTauE_3$ have the same projection 
in the transverse tree $\Es_{\p_1}$,
that is the center of the ideal triple
$ (\p_1\p_{13}, \p_1\p_2, \p_1\p_3)
= (\D_1, \p_1\p_2, \p_1\p_3)
= (\p_1\p_{23}, \p_1\p_2, \p_1\p_3)
$,
proving that $\rac2(\ovra{\yTauE_2\yTauE_3})=0$.
%
Furthermore, 
%
by Proposition \ref{prop- 1 plat, 2 points} 
%
we have
\begin{align*}
  \rac2(\ovra{\yTauE_3 \yTauE_1})
  &= \geombir(\p_1\p_2, \p_1\p_{12},  \p_1\p_3 , \p_1\p_{23} )\\
  &= \geombir(\p_1\p_2, \D_1,  \p_1\p_3 , \p_1\p_{23} )\\
  &= \geombir(\D_1, \p_1\p_2, \p_1\p_{23}, \p_1\p_3)\\
  &=\gZo
\end{align*}
proving that $\rac2(\ovra{\yTauE_3\yTauE_1})=\gZo$.
%
Applying this to the permuted triple $(\F_3,\F_1,\F_2)$, we obtain
$\rac1(\ovra{\yTauE_2\yTauE_3}) = \gZo$ (by invariance of the
geometric triple
ratio $\gZo$ by cyclic permutation).
%
Assertion (\ref{it- y_i y_i+1})
follows Assertion (\ref{it- y_i y_i+1}), applying cyclic permutations.
%
\end{proof}







We now describe the intersections of $\Ap$ with the other flats, 
see figure \ref{fig-  in Ap}.

\begin{figure}[h]

%
  \begin{minipage}[b]{0.4\linewidth}
\centering
%
\includegraphics{figures/Es_Ap_triangle_pos_yTauE_avec_gZ.fig}
%
    
In the case $\gZo\geq 0$.
  \end{minipage}
%
\hfill
%
\begin{minipage}[b]{0.4\linewidth}
\centering
%
\includegraphics{figures/Es_Ap_triangle_neg_yTauE_avec_gZ.fig}
%
    
In the case $\gZo\leq 0$.
  \end{minipage}
%

\caption{In the flat $\Ap$.}
\label{fig-  in Ap}
\end{figure}
%
\begin{prop}
\label{prop- in Ap - intersections}
%
Let $\SW_\ii=\Ap \cap  \App_{\ii,\ii+1}$ 
and let $\Delta$ be the 
%
triangle with vertices $\yTauE_1,\yTauE_2,\yTauE_3$.
Then

\begin{enumerate}
\item 
\label{i- Ap cap Aij}
$\SW_\ii$ is the \sectorE{} of $\Ap$ bounded by the rays 
from $\yTauE_{\ii+2}$ to $\p_\ii$ and $\p_{\ii+1}$.
%


\item 
\label{i- partition Ap}
$\SW_1,\SW_2,\SW_3$ and $\Delta$ form a partition of $\Ap$.

%

%

%

\end{enumerate}
%
\end{prop}


\begin{proof}
Assertion (\ref{i- Ap cap Aij}) follows from
point (\ref{i- Aij cap Ap}) 
of Proposition \ref{prop- in Aij - intersections}.
%
In the case where $\gZo\geq 0$, 
Assertion (\ref{i- partition Ap}) then comes 
from the fact that 
for all $\ii$, $\yTauE_{\i+1}$ is in the ray $[\yTauE_\ii,\p_{\ii+2})$ 
(Proposition \ref{prop- in Ap}).
%
The case where $\gZo\leq 0$ is similar.
%
\end{proof}


\subsection{In the flat \texorpdfstring{$\AD$}{A\_D}}
%
In the dual  flat $\AD=\App(\D_1,\D_2,\D_3)$,
we have   properties to the case of flat $\Ap$, 
which have similar proofs
(exchanging the role of points and lines).

%

\begin{prop}
\label{prop- in AD}
%
In the flat $\AD$  identified with $\Aa$ by a marked flat 
sending $\bord\Cc$ to $(\D_1\cap\D_2,\D_1)$, we have:
%
\begin{enumerate}
\item   $\ovra{\yTauE^*_2\yTauE^*_3}=(0,-\gZo)$
in simple roots coordinates.
%
In particular $\yTauE^*_2$ and $\yTauE^*_3$ 
are on a common singular geodesic to $\D_1$.
%
\item The points  $\yTauE^*_1$, $\yTauE^*_2$,  $\yTauE^*_3$ 
form a singular triangle $\Delta^*$ in $\AD$.

\item 
\label{i- AD cap Aij} For all $\ii\in\ZZ/3\ZZ$,
$\SW^*_\ii=\AD \cap  \App_{\ii,\ii+1}$  is the \sectorE{} of $\AD$ 
bounded by the rays from $\yTauE^*_{\ii+2}$ to $\D_\ii$ and $\D_{\ii+1}$.

\item 
\label{i- partition AD}
$\SW^*_1,\SW^*_2,\SW^*_3$ and $\Delta^*$ form a partition of $\AD$.

\end{enumerate}
%
\qed
\end{prop}


\begin{figure}[h]

%
  \begin{minipage}[b]{0.4\linewidth}
\centering
%
\includegraphics{figures/Es_AD_triangle_pos_yTauE.fig}
    
In the case $\gZo\geq 0$.
  \end{minipage}
%
\hfill
%
  \begin{minipage}[b]{0.4\linewidth}
\centering
%
\includegraphics{figures/Es_AD_triangle_neg_yTauE.fig}
    
In the case $\gZo\leq 0$.
  \end{minipage}
%

\caption{In the flat $\AD$.}
\label{fig-  in AD}
\end{figure}


\subsection{Classification}
We now combine the previous results  
to establish the classification in two geometric types, 
finishing to prove Theorems \ref{theo- tripod} and \ref{theo- triangle}.


\begin{proof}[Proof of Theorem \ref{theo- tripod}]
Let $\vstE=\yTauE_3$ and $\vstE^*=\yTauE_3^*$.
%
We identify the flat $\App_{12}$
with the model flat $\Aa$ 
by a marked flat sending $\bord\Cc$ to $\F_2$, and $0$ to 
$\yTauE_3^*$.
%
By proposition \ref{prop- in Aij - intersections}
applied to the flat $\App_{12}$, we have
$\rac1(\yTauE_3)=\gZp$,
$\rac2(\yTauE_3)=\gZpp$, and
$\rac3(\yTauE_3)=\gZo$.
%
By proposition \ref{prop- in Aij - intersections}
applied to the flat $\App_{12}$,
the intersection $I=\App_{12}\cap \App_{23} \cap \App_{31}$ 
is the  subset of $\yE\in \App_{12}$ such that
$$
\left\{
\begin{array}{lll}
  0 &\leq \rac1(\yE) &\leq \rac1(\yTauE_3)=\gZp\\  
  0 &\geq \rac2(\yE) &\geq  \rac2(\yTauE_3)=\gZpp\\ 
  \max(\rac3(\yTauE_3), 0)& 
  \leq \rac3(\yE)  
  &\leq  \min(\rac3(\yTauE_3), 0) 
  \; .
\end{array}
\right.
$$
%
In particular, 
if $I$ is not empty, then
$\gZo=\rac3(\yTauE_3)=0$.

Suppose from now on that $\gZo=0$. 
%
Then  $\gZp\geq 0$ and $\gZpp =-\gZp$  by 
the ultrametricity of the geometric triple ratio
(Proposition \ref{prop- props of geomtri}, (\ref{item- geomtri is ultram})).
By the description above, $I$ 
is then the subset of the line $\rac3=0$ (which contains
$\yTauE_3^*=0$ and $\yTauE_3$) consisting of the $\yE$ such that
$ 0 \leq \rac1(\yE) \leq \rac1(\yTauE_3)$
(since $\rac2(\yE)=-\rac1(\yE)$ when $\rac3(\yE)=0$).
%
Hence $I$ is  not empty and 
is the segment from $0=\yTauE_3^*$ to $\yTauE_3$ 
i.e. $[\vstE,\vstE^*]$.
%
Furthermore, 
as $\gZo=0$,  Proposition \ref{prop- in Ap}  implies that 
%
$\yTauE_1=\yTauE_2=\yTauE_3$. 
%
Similarly, 
we have  $\yTauE_1^*=\yTauE_2^*=\yTauE_3^*$ 
by Proposition \ref{prop- in AD}.
%
Suppose now $\vstE\neq \vstE^*$.
Since the segment $[\vstE,\vstE^*]$ lies in the ray
$[\vstE,\p_\iijj)$,
and $\vstE=\yTauE_\kk$ is the orthogonal projection of
$\p_\iijj$ on $\Ap$, 
we have $\angle_\vstE(\vstE^*,\D)=\pi$ 
for all lines $\D$ in $\bordinf\Ap$ 
(Proposition \ref{prop- center of a projective frame}).
Therefore  we have
%
$\angle_\vstE(\vstE^*,\yE)\geq \frac{2\pi}{3}$
for all $\yE\neq\vstE$ in $\Ap$.
%
Similarly,  we have that $\angle_{\vstE^*}(\vstE,\yE)\geq
\frac{2\pi}{3}$ for all $\yE\neq\vstE$ in $\Ap$.
%
Hence  $[\vstE,\vstE^*]$ is the unique segment of minimal length joining
$\Ap$ to $\AD$.
%
Assertion (\ref{i- coords xx^*}) 
follows from Proposition \ref{prop- in  Aij - y*y}.
%
\end{proof}

















\begin{proof}[Proof of Theorem \ref{theo- triangle}]
%
If $\gZp > 0$, then  $\gZo = 0$
by the ultrametricity of the geometric triple ratio
(Proposition \ref{prop- props of geomtri}(\ref{item- geomtri is ultram})), 
and  $\Ap \cap \AD$ is empty by 
Theorem \ref{theo- tripod}.
%
Suppose now that $\gZp\leq 0$. 
%
Since the case $\gZo \leq 0 $ reduces to the case $\gZo \geq 0 $ by 
exchanging $\F_2$ and $\F_3$,
%
it is enough to handle the case $\gZo \geq 0$. 
%
Then $\gZpp=0$ and $\gZp=-\gZo$. 
%
Let $\vstE_\ii=\yTauE_{\ii+2}$ for $\ii\in\ZZ/3\ZZ$.
%
In $\Aij$ identified with $\Aa$ in such a way that $\yTauE_\kk^*=0$,
by Proposition \ref{prop- in Aij - y*y} we have 
$\rac1(\yTauE_\kk)=\gZp=-\gZo \leq 0$,
 $\rac2(\yTauE_\kk)=\gZpp = 0$,
hence $\rac3(\yTauE_\kk)=\gZo \geq 0$.
%
By assertion (\ref{i- Aij cap Aki}) 
of Proposition \ref{prop- in Aij -  intersections},
$\Aij\cap\Aik$ is the set of $\xE\in \Aij\simeq\Aa$
such that
$\rac1(\xE)\leq   \rac1(\yTauE_\kk)$,
$\rac2(\xE)  \leq  0=\rac2(\yTauE_\kk)$
and $\rac3(\xE)  \geq  \max(\rac3(\yTauE_\kk), 0)=\rac3(\yTauE_\kk)$.
%
This is the Weyl chamber $\yTauE_\kk-\Cb$, 
i.e. the Weyl chamber from  $\yTauE_\kk=\vstE_\ii$ to $\F_\ii$.
%
Similarly, $\Aij\cap\Ajk$ 
is the Weyl chamber from  $\yTauE_\kk^*$ to $\F_\jj$.
%
Applying a cyclic permutation $(\ii \jj \kk)$,
i.e. working in the flat $\Ajk$, 
we also similarly  get that $\Aij\cap\Ajk$ 
is the Weyl chamber from  $\yTauE_\ii$ to $\F_\jj$.
Therefore $\yTauE_\kk^*=\yTauE_\ii$.
%

By Proposition \ref{prop- in Aij - intersections}
$\Ap \cap \AD \cap \Aij$ 
is the intersection 
of the \sectorE{} at $\yTauE_\kk^*$  
bounded by the rays to $\D_\ii$ and $\D_\jj$,
with the \sectorE{}  at $\yTauE_\kk$  
bounded by the rays to $\p_\ii$ and $\p_\jj$.
As the point $\yTauE_\kk$ is on the ray from $\yTauE_\kk$ to $\D_\ii$, 
this is equal to the segment $[\yTauE_\kk,\yTauE_\kk^*]$. 
%
In particular $\Ap \cap \AD $ contains $\yTauE_\kk$.
%
Then
$\Ap \cap \AD$ contains $\yTauE_1$, $\yTauE_2$ and $\yTauE_3$, hence
the triangle $\Delta$ with vertices $\yTauE_1$, $\yTauE_2$ and $\yTauE_3$, 
and  
since 
$\Ap \cap \AD \cap \Aij 
= [\yTauE_\kk,\yTauE_\ii]
\subset \Delta$, 
the  assertion (\ref{i- partition Ap}) 
of Proposition \ref{prop- in  Ap - intersections} 
provides the reverse inclusion. 
%
Assertion (\ref{i- xi xj}) comes from Proposition 
\ref{prop- in Aij -  y*y}.
%

We finally prove assertion (\ref{i- Delta opp Fi en xi}). 
Let $(\ii,\jj,\kk)=(1 2 3)$.
Looking in the flat $\Ap$, we see that 
the singular triangle $\Delta$ is contained in the Weyl chamber of $\Es$
with tip $\vstE_\ii$  and that
at $\vstE_\ii$, we have
$\TangS_{\vstE_\ii} \vstE_\jj = \TangS_{\vstE_\ii} \p_\jj$.
Looking in the flat $\AD$ we get
$\TangS_{\vstE_\ii} \vstE_\kk = \TangS_{\vstE_\ii} \D_\kk$. 
%
Hence $\TangS_{\vstE_\ii} \Delta=(\TangS_{\vstE_\ii}\p_\jj,\TangS_{\vstE_\ii} \D_\kk)$.
%
Since
$\vstE_\ii$ belongs to the flats $\App(\F_\ii,\F_\jj)$ and
$\App(\F_\ii,\F_\kk)$, 
we have that
$\TangS_{\vstE_\ii} \p_\jj$ is opposite to $\TangS_{\vstE_\ii} \D_\ii$
and that
$\TangS_{\vstE_\ii} \D_\kk$ is opposite to $\TangS_{\vstE_\ii} \p_\ii$.
%
Therefore the Weyl chambers
$\TangS_{\vstE_\ii} \Delta$ and $\TangS_{\vstE_\ii} \F_\ii$ are opposite.
%
It implies that $\Delta$ and the Weyl chamber from $\vstE_\ii$ to
$\F_\ii$ are contained in a common flat of $\Es$ by basic properties of
real Euclidean buildings (see property (CO) of
\cite{ParImm}).
%
\end{proof}























In the algebraic case the following remark 
provides an alternative proof of some of the assertions 
of  Theorem  \ref{theo- triangle}.
%
\begin{rema}
Let $\pt_\ii$ in $\V=\KK^3$ be a vector  representing $\p_\ii$ and
$\Dt_\ii$ in $\V^*$ be a linear form  representing $\D_\ii$.
%
Let $\eVb=(\eV_1,\eV_2,\eV_3)$ 
be the basis of $\V$ dual to the basis
$(\Dt_1,\Dt_2,\Dt_3)$ of $\V^*$. 
%
Then in the projective plane $\classPV{\eV_\ii}=\D_\jj\cap \D_\kk$.
%
We may suppose that $\pt_1=(0,1,1)$, $\pt_2=(\Z,0,1)$, $\pt_3=(1,1,0)$
in the basis $\eVb$, with $\Z=\Tri(\F_1,\F_2,\F_3)$.
%
Then the element $\g\in\GL(V)$ with matrix in the basis $\eVb$ 
$$
\begin{pmatrix}
1   & 1 & 0\\
0   & 1 & 1 \\
1/\Z & 0 & 1
\end{pmatrix}$$
%
sends $\classPV{\eV_\ii}$ to $\p_{\ii+1}$, 
hence $\AD$ to $\Ap$.
%
If $\abs{1+\Z}\geq 1$ and
%
$\gZ=\log\abs{\Z}\geq 0$, 
then the  fixed point set  of $\g$ in $\AD$ 
is the image by the marked flat $\f_\eVb$
of the singular triangle 
$\{\vA\in\Cb \ | \ \vA_1-\vA_3 \leq \log\abs{\Z}\}$ (that is, $\Delta$).
%
\end{rema}




\subsection{Complements}
We add here for future use  a simple description
 of the vertices $\vstE_\ii,\vstE_\jj,\vstE_\kk$ of the singular triangle 
%

in Theorem \ref{theo- triangle}
by the projections on transverse trees at infinity.

\begin{lemm}
\label{lemm- proj de x_i sur les arbres transverses}
Under the hypotheses and notations of Theorem \ref{theo- triangle},
in the transverse trees at infinity $\Es_{\p_\i}$ and $\Es_{\D_\i}$ we have
\begin{enumerate}
\item 
\label{i- proj de xEi sur tree at p_i}
The projection $\projE_{\p_\i}(\vstE_\ii)$ of $\vstE_\ii$ on $\Es_{\p_\i}$  
%
is  the center of the ideal tripod $\D_\i$, $\p_\i\p_\j$, $\p_\i\p_\k$.
%

\item 
\label{i- proj de xEi sur tree at D_i}
The projection $\projE_{\D_\i}(\vstE_\ii)$ of $\vstE_\ii$  on $\Es_{\D_\i}$
is  the center  of the ideal tripod $\p_\i$, $\D_\i\cap\D_\j$, $\D_\i\cap\D_\k$.
%

\item
\label{i- proj de xEj sur tree at p_i}
 The projection $\projE_{\p_\i}(\vstE_\jj)$ 
%
is  the center of the ideal tripod 
$\D_\i$, $\p_\i\p_\j$, $\p_\i\p_\jk$.

\item
\label{i- proj de xEj sur tree at D_i}
 The projection  $\projE_{\D_\i}(\vstE_\jj)$ 
%
is  the center of the ideal tripod 
$\p_\i$, $\D_\i\cap\D_\j$, $\D_\i\cap \D_\jk$.
\end{enumerate}

\end{lemm}


\begin{proof}
%
As $\vstE_\ii$ belongs to the three flats 
$\App(\F_\k,\F_\i)$ and $\App(\F_\j,\F_\i)$ and $\App(\p_\i,\p_j,\p_k)$,
its projection in the tree $\Es_{\p_\i}$
belongs to the projection of $\App(\F_\j,\F_\i)$, 
which is the line from $\D_\i$ to $\p_\i\p_\j$,
to the projection of $\App(\F_\k,\F_\i)$, 
which is the line from $\D_\i$ to $\p_\i\p_\k$,
and to the projection of $\App(\p_\i,\p_\j,\p_k)$, 
which is the line from $\p_\i\p_j$ to $\p_\i\p_\k$.
Hence (\ref{i- proj de xEi sur tree at p_i}) is proven.
%
The statement (\ref{i- proj de xEi sur tree at D_i})
%
is proven in the same way.

We now prove (\ref{i- proj de xEj sur tree at p_i}).
By (\ref{i- proj de xEi sur tree at D_i}) applied to $\vstE_\jj$, 
we have that $\projE_{\D_\j}(\vstE_\jj)$
is  the center  of the ideal tripod 
$\p_\j$, $\p_\jk=\D_\j\cap\D_\k$,$\D_\j\cap\D_\i$. 
As $\vstE_\jj$ is on a geodesic from $\D_\j$ to $\p_\i$, 
we may deduce that $\projE_{\p_\i}(\vstE_\jj)$
is  the center  of the ideal tripod 
$\p_\i\p_\j$, $\p_\i\p_\jk$, $\D_\i$ 
(using the canonical isomorphim $\Es_{\D_\j}\isomto \Es_{\p_\i}$).
%
The last statement (\ref{i- proj de xEj sur tree at D_i})
%
has identical proof.
%
\end{proof}









triples_4_bibli.tex
%



\begin{thebibliography}{Otal92}
%

\bibitem[Bal08]{Balser08}
A. ~Balser,
{\em Convex rank 1 subsets of Euclidean buildings (of type A2)},
Geom. Dedicata {\bfseries 131} (2008), 123–158.

%

\bibitem[Bou96]{Bourdon96}
M. Bourdon, {\em Sur le birapport au bord des CAT(-1)-espaces},
Pub. Math. I.H.E.S. {\bfseries 83} (1996), 95 – 104. 

\bibitem[BrHa99]{BrHa99}
M.R.~Bridson, A.~Haefliger, {\em Metric spaces with non-positive curvature}, 
Grund.~math.~Wiss. {\bfseries 319}, Springer Verlag, 1999.

%
\bibitem[BrTi84]{BrTi84} F.~Bruhat, J.~Tits, {\em Schémas en groupes et
immeubles des groupes classiques sur un corps local},
Bull. Soc. Math. Fr. {\bfseries 112} (1984), 259-301.


%
\bibitem[BIW10]{BIW10}
M.~Burger, A.~Iozzi, A.~Wienhard,
{\em Surface group representations with maximal Toledo invariant},
Ann. of Math. {\bfseries 172} (2010), 517-566. 


%

%
\bibitem[FoGo06]{FoGoIHES}
 V.~V.~Fock, A.~B.~Goncharov,
{\em Moduli spaces of local systems and higher Teichmüller theory}
Publ. Math. IHES. {\bfseries 103} (2006), pp 1–211.

%
\bibitem[FoGo07]{FoGoSPC}
 V.~V.~Fock, A.~B.~Goncharov, {\em Moduli spaces of convex projective
 structures on surfaces}, Adv. Math. {\bfseries 208} (2007),  pp249–273.

%
\bibitem[GoIw63]{GoIw63}
O.~Goldman, N.~Iwahori, {\em The space of $p$-adic norms}, 
Acta Math. {\bfseries 109} (1963), 137-177.

%

%
\bibitem[Go80]{Gol80}
W. M. Goldman,
{\em Discontinuous groups and the Euler class}, 
Thesis (Ph.D.) - University of California, Berkeley (1980).

%
\bibitem[Go99]{GolBook}
W.~M.~Goldman, 
{\em Complex hyperbolic geometry},
 Oxford Mathematical Monographs (1999). 

%

%

\bibitem[KlLe97]{KlLe97}
B.~Kleiner, B.~Leeb,  {\em Rigidity of quasi-isometries for symmetric 
spaces of higher rank}, 
Pub. Math. IHES {\bfseries 86} (1997) 115-197.

\bibitem[Leeb00]{Leeb00}
B.~Leeb, {\em A characterization of irreducible symmetric spaces 
and Euclidean buildings of higher rank by their asymptotic geometry},
Bonn. Math. Schr. 326,  Universität Bonn, Mathematisches Institut,
Bonn,  2000. 

\bibitem[MSVM14]{MSVM14-I} 
B.~Mühlherr, K.~Struyve, H.~Van Maldeghem, 
{\em Descent of affine buildings - I. Large minimal
angles}, Trans. Amer. Math. Soc. {\bfseries 366} (2014),  4345–4366. 

%


\bibitem[Otal92]{Otal92}
J.-P.~Otal, {\em Sur la géometrie symplectique de l'espace des
  géodésiques d'une variété à courbure négative},
Rev. Mat. Iberoam. {\bfseries 8} (1992), 441–456.

\bibitem[Par99]{ParImm}
A. Parreau {\em Immeubles affines, construction par les normes et étude des
isométries}, Crystallographic groups and their generalizations
(Kortrijk, 1999), 263--302, Contemp. Math., 262, Amer. Math. Soc.,
2000.


%

\bibitem[Par15]{ParDegFG}
 A. Parreau, 
{\em Invariant subspaces for some surface groups 
acting on $\typeA_2$-euclidean buildings},
preprint.

%

\bibitem[Rou09]{Rousseau09} 
G.~Rousseau, 
{\em Euclidean buildings},
In: 
{\em Géométries à courbure négative ou nulle, groupes discrets et
  rigidités}, 
Sémin. Congr. {\bfseries 18} (2009), 77–116. 


\bibitem[Tal01]{Tal01Th}
M. Talbi, {\em Inégalité de Haagerup et géométrie des groupes}, 
Ph.D. thesis, Univ. Lyon I, 2001.

%

%

\bibitem[Tits74]{Tits74}
J.~Tits, {\em Buildings of spherical type and finite BN-pairs},
Lectures Notes in Math. {\bfseries 386}, Springer-Verlag, 1974. 

\bibitem[Tits86]{Tits86}
J.~Tits, {\em Immeubles de type affine},
dans ``Buildings and the geometry of diagrams'', Proc.~CIME Como 1984,
  L.~Rosati ed., Lect. Notes {\bfseries 1181}, Springer Verlag, 1986, 159-190.

\bibitem[Tol89]{Tol89}
D.~Toledo, 
{\em Representations of surface groups in complex hyperbolic space},
J. Diff. Geom. {\bfseries 29} (1989).

%

\end{thebibliography}

1504.00285.extracted_bib
[Bal08]{Balser08 A. ~Balser, {\em Convex rank 1 subsets of Euclidean buildings (of type A2), Geom. Dedicata {series 131 (2008), 123–158. %
1504.00285.extracted_bib
[Bou96]{Bourdon96 M. Bourdon, {\em Sur le birapport au bord des CAT(-1)-espaces, Pub. Math. I.H.E.S. {series 83 (1996), 95 – 104.
1504.00285.extracted_bib
[BrHa99]{BrHa99 M.R.~Bridson, A.~Haefliger, {\em Metric spaces with non-positive curvature, Grund.~math.~Wiss. {series 319, Springer Verlag, 1999. %
1504.00285.extracted_bib
[BrTi84]{BrTi84 F.~Bruhat, J.~Tits, {\em Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. Fr. {series 112 (1984), 259-301. %
1504.00285.extracted_bib
[BIW10]{BIW10 M.~Burger, A.~Iozzi, A.~Wienhard, {\em Surface group representations with maximal Toledo invariant, Ann. of Math. {series 172 (2010), 517-566. % %
1504.00285.extracted_bib
[FoGo06]{FoGoIHES V.~V.~Fock, A.~B.~Goncharov, {\em Moduli spaces of local systems and higher Teichmüller theory Publ. Math. IHES. {series 103 (2006), pp 1–211. %
1504.00285.extracted_bib
[FoGo07]{FoGoSPC V.~V.~Fock, A.~B.~Goncharov, {\em Moduli spaces of convex projective structures on surfaces, Adv. Math. {series 208 (2007), pp249–273. %
1504.00285.extracted_bib
[GoIw63]{GoIw63 O.~Goldman, N.~Iwahori, {\em The space of $p$-adic norms, Acta Math. {series 109 (1963), 137-177. % %
1504.00285.extracted_bib
[Go80]{Gol80 W. M. Goldman, {\em Discontinuous groups and the Euler class, Thesis (Ph.D.) - University of California, Berkeley (1980). %
1504.00285.extracted_bib
[Go99]{GolBook W.~M.~Goldman, {\em Complex hyperbolic geometry, Oxford Mathematical Monographs (1999). % %
1504.00285.extracted_bib
[KlLe97]{KlLe97 B.~Kleiner, B.~Leeb, {\em Rigidity of quasi-isometries for symmetric spaces of higher rank, Pub. Math. IHES {series 86 (1997) 115-197.
1504.00285.extracted_bib
[Leeb00]{Leeb00 B.~Leeb, {\em A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonn. Math. Schr. 326, Universität Bonn, Mathematisches Institut, Bonn, 2000.
1504.00285.extracted_bib
[MSVM14]{MSVM14-I B.~Mühlherr, K.~Struyve, H.~Van Maldeghem, {\em Descent of affine buildings - I. Large minimal angles, Trans. Amer. Math. Soc. {series 366 (2014), 4345–4366. %
1504.00285.extracted_bib
[Otal92]{Otal92 J.-P.~Otal, {\em Sur la géometrie symplectique de l'espace des géodésiques d'une variété à courbure négative, Rev. Mat. Iberoam. {series 8 (1992), 441–456.
1504.00285.extracted_bib
[Par99]{ParImm A. Parreau {\em Immeubles affines, construction par les normes et étude des isométries, Crystallographic groups and their generalizations (Kortrijk, 1999), 263--302, Contemp. Math., 262, Amer. Math. Soc., 2000. %
1504.00285.extracted_bib
[Par15]{ParDegFG A. Parreau, {\em Invariant subspaces for some surface groups acting on $\typeA_2$-euclidean buildings, preprint. %
1504.00285.extracted_bib
[Rou09]{Rousseau09 G.~Rousseau, {\em Euclidean buildings, In: {\em Géométries à courbure négative ou nulle, groupes discrets et rigidités, Sémin. Congr. {series 18 (2009), 77–116.
1504.00285.extracted_bib
[Tal01]{Tal01Th M. Talbi, {\em Inégalité de Haagerup et géométrie des groupes, Ph.D. thesis, Univ. Lyon I, 2001. % %
1504.00285.extracted_bib
[Tits74]{Tits74 J.~Tits, {\em Buildings of spherical type and finite BN-pairs, Lectures Notes in Math. {series 386, Springer-Verlag, 1974.
1504.00285.extracted_bib
[Tits86]{Tits86 J.~Tits, {\em Immeubles de type affine, dans ``Buildings and the geometry of diagrams'', Proc.~CIME Como 1984, L.~Rosati ed., Lect. Notes {series 1181, Springer Verlag, 1986, 159-190.
1504.00285.extracted_bib
[Tol89]{Tol89 D.~Toledo, {\em Representations of surface groups in complex hyperbolic space, J. Diff. Geom. {series 29 (1989). %