Coverings in $p$-adic analytic geometry and log coverings I:\\ Cospecialization of the $(p')$-tempered fundamental group for a family of curves
Emmanuel Lepage
The tempered fundamental group of a $p$-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite \'etale covering of the space. Here we construct cospecialization homomorphisms between $(p')$ versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction and we will prove the invariance of the geometric log fundamental group of log smooth log schemes over a log point by change of log point.
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\title%[Cospecialization of tempered groups for curves]
{Coverings in $p$-adic analytic geometry and log coverings I:\\
Cospecialization of the $(p')$-tempered fundamental group for a family of curves}
\author{Emmanuel Lepage}
%\email{emmanuel.lepage@ens.fr}
%\address{D.M.A., E.N.S, 45 rue d'Ulm, 75005 PARIS, FRANCE}
%\classification{11G20,14H30,14G22}
%\keywords{fundamental groups, Berkovich spaces, specialization}

\maketitle
%\tableofcontents
\begin{abstract}
The tempered fundamental group of a $p$-adic analytic space classifies
coverings that are dominated by a topological covering (for the Berkovich
topology) of a finite \'etale covering of the space. Here we construct
cospecialization homomorphisms between $(p')$ versions of the tempered fundamental
groups of the fibers of a smooth family of curves with semistable reduction. To
do so, we will translate our problem in terms of cospecialization morphisms of
fundamental groups of the log fibers of the log reduction and we
will prove the invariance of the geometric log fundamental group of
log smooth log schemes over a log point by change of log
point.
\end{abstract}
\section*{Introduction}
 In general topology, the fundamental group  
of a connected locally contractible space classifies its (unramified)  
coverings. A. Grothendieck developed an avatar in abstract algebraic  
geometry: he attached to any algebraic variety a profinite  
fundamental group, which classifies its finite \'etale coverings. For
a complex algebraic variety, Grothendieck's fundamental group is
canonically isomorphic to the profinite completion of the topologic
fundamental group of the corresponding topological space.\\  
Here we are interested in an analog in $p$-adic geometry. More precisely we
will study the \emph{tempered fundamental group} of $p$-adic varieties
defined by Y. André.  The profinite completion of the tempered fundamental
group of any smooth $p$-adic algebraic variety coincides with Grothendieck's algebraic
fundamental group. It also accounts for the usual (infinite) ``uniformizations'' in
$p$-adic analytic geometry such as the uniformization of Tate elliptic curves,
which are historically at the very basis of $p$-adic rigid
geometry. Such uniformizations give infinite discrete quotients
of the tempered fundamental group.\\
Since the analytification (in the sense of V. Berkovich or of rigid geometry)
of a finite étale covering of a $p$-adic algebraic variety is not
necessarily a topological covering, André had to consider a category of
coverings slightly bigger than just the category of topological coverings.
He defined tempered coverings, which are (possibly infinite) \'etale coverings in
the sense of A.J. de Jong (that is to say, which are,  
Berkovich-locally on the base, direct sums of finite coverings)
 such that, after pulling
back by some finite \'etale covering, they become topological coverings (for the
Berkovich topology). The \emph{tempered fundamental group} is the prodiscrete group that classifies those tempered coverings.  To give a more handful description, if
one has a sequence of pointed finite Galois connected coverings $((S_i,s_i))_{i\in \mbf N}$ such that the corresponding pointed pro-covering of $(X,x)$ is the
universal pro-covering of $(X,x)$, and if $(S^{\infty}_i,s^{\infty}_i)$ is a
universal topological covering of $S_i$, the tempered fundamental group of
$X$ can be seen as $\gtemp(X,x)=\varprojlim_i \Gal(S^{\infty}_i/X)$.
Therefore, to understand the tempered fundamental group of a variety, one
has to understand the topological behavior of its finite \'etale
coverings.\\
In the case of a curve, the question becomes more concrete since there
is a natural embedding of the geometric realization of the graph of its
stable model into the Berkovich space of the curve which is a homotopy
equivalence.\\
Among applications of tempered fundamental groups, let us cite in passing
the theory of $p$-adic orbifolds and $p$-adic triangle groups \cite{andre1}
and a
$p$-adic version of Grothendieck-Teichm\"uller theory \cite{andre2}.\\

In this article we will be interested in the variation of the tempered
fundamental group of the fibers of a family of curves. This article will be
followed by another one~\cite{cosp2}, in which we will consider higher dimensional
families.\\
For a proper morphism of schemes $f: X\to Y$ with geometrically connected
fibers and a specialization $ 
\bar y_1 \to \bar y_2$ of geometric points of $Y$, A. Grothendieck  
has defined algebraic fundamental groups $\pi_1^{alg}(X_{\bar y_i})$  
and a specialization homomorphism  $\pi_1^{alg}(X_{\bar y_1})\to  
\pi_1^{alg}(X_{\bar y_2})$. Grothendieck's specialization theorem 
tells that this homomorphism is surjective if $f$ is separable and induces an  
isomorphism between the prime-to-$p$ quotients if $f$ is smooth  
(here, $p$ denotes the characteristic of $\bar y_2$), cf. \cite[cor. X.2.4,
cor. X.3.9]{sga}.\\
In complex analytic geometry, a smooth and proper morphism is locally a trivial
fibration of real differential manifolds, so that, in particular, all the
fibers are homeomorphic, and thus have isomorphic (topological) fundamental
groups.\\
The aim of this paper is to find some analog of the specialization
theorem of Grothendieck in the case of the tempered fundamental group.\\
In this paper, we will concentrate on the case of curves.\\

One problem which appears at once in looking for some non archimedean analog of
Grothendieck's specialization theorems is that there are in general no non trivial
specializations between distinct points of a non archimedean analytic
(Berkovich or rigid) space: for example a separated Berkovich space has a
Hausdorff underlying topological space, so that if there is a
cospecialization (for the Berkovich topology, the étale topology\dots)
between two geometric points of a Berkovich space, the two geometric points
must have the same underlying point. Thus we will assume we have a
model over the ring of integers of our non-archimedean field (with good enough properties) and we will look at
the specializations in the special fiber.\\
 
We want to understand how the tempered fundamental group of the geometric fibers of a
smooth family varies. Let us for instance consider a family of elliptic curves. The tempered
fundamental group of an elliptic curve over a complete algebraically closed
non archimedean closed field is $\widehat{\mbf Z}^2$ if it has
good reduction, and $\widehat{\mbf Z}\times \mbf Z$ if it is a Tate
curve. In particular, by looking at a moduli space of stable pointed elliptic curves
with level structure\footnote{to avoid stacks. However, the cospecialization
  homomorphisms we will construct will be local for the étale topology of
  the special fiber of the base. Thus, the fact that the base is a
  Deligne-Mumford stack is not really a problem.}, the tempered fundamental group (or any reasonable
$(p')$-version) cannot be constant.\\
Moreover, if one looks at the moduli space over $\mbf Z_p$, and 
considers a curve $E_1$ with bad reduction and a curve $E_0$ with generic
reduction (hence good reduction), there cannot be a morphism
$\gtemp(E_0)\to\gtemp(E_1)$ which induces Grothendieck's specialization on
the profinite completion, although the reduction point corresponding to
$E_1$ specializes to the reduction point corresponding to $E_0$. Therefore
there cannot be any reasonable specialization theory.\\
On the contrary, if
one has two geometric points $\eta_1$ and $\eta_2$ of the moduli space such
that the reduction of $\eta_1$ specializes to the reduction of $\eta_2$,
then $E_{\eta_1}$ has necessarily better reduction than $E_{\eta_2}$ and
there is some morphism
$\gtemp(E_{\eta_2})\to\gtemp(E_{\eta_1})$ that induces an isomorphism between
the profinite completions. Thus we want to look for a \emph{cospecialization} of the tempered fundamental group.\\
 
The topological behavior of general finite étale coverings is too
complicated to hope to have a simple cospecialization theory without
adding a $(p')$ condition on the coverings: for example
two Mumford curves over some finite extension of $\mbf Q_p$ with isomorphic
geometric tempered fundamental group have the same metrized graph of stable reduction. Thus even if two Mumford curves
have isomorphic stable reduction (and thus the point corresponding to their stable reduction is the same), they may not have isomorphic
tempered fundamental group in general. Thus we will only
study here finite coverings that are dominated by a finite Galois covering
whose order is prime to $p$, where $p$ is the residual characteristic
(which can be $0$; such
a covering will be called a $(p')$-finite covering). Then, it becomes
natural to introduce a $(p')$-tempered fundamental group which classifies
tempered coverings that become topological coverings after pullback along
some $(p')$-finite covering. It should be remarked that this
$(p')$-tempered fundamental group cannot be in general recovered from the
tempered fundamental group: for example, if $p=2$, $\gtemp(X)^{(p')}=\{1\}$ for an Enriques surface over $\mbf C_p$ with good reduction and $\gtemp(X)^{(p')}=\mbf Z/2\mbf Z$ for an Enriques surface with semistable reduction whose dual simplicial complex is homotopic to a real projective plane, whereas, in both cases, $\gtemp(X)=\mbf Z/2\mbf Z$. However, in the case of a curve, one can recover the $(p')$-tempered fundamental group from the tempered fundamental group.\\
The $(p')$-tempered fundamental group of a curve was already studied by
S. Mochizuki in~\cite{mochi}. It can be described in terms of a graph of profinite
groups. From this description, one easily sees that the isomorphism class
of the $(p')$-tempered
fundamental group of a $p$-adic curve depends only of the stratum of the
Knudsen stratification of the moduli space of stable curves in which the
stable reduction lies. Moreover if one has two strata $x_1$ and $x_2$ in the moduli space
of stable curves such that $x_1$ is in the closure of $x_2$, one can easily
construct  morphisms from the graph of groups
corresponding to $x_1$ to the graph of groups corresponding to $x_2$
(inducing morphisms of tempered fundamental groups which induce 
isomorphisms of the pro-$(p')$ completions).\\



\subsection*{}
We shall study the following situation. Let $O_K$ be a complete discrete
valuation ring, $K$ be its fraction field, $k$ be its residue field and $p$ be
its characteristic (which can be $0$).
Let $X$ be a proper semistable pointed curve over $O_K$ smooth over $K$ and let $U\subset
X_{\eta}$ be the complement of the marked points. Let us describe the tempered
fundamental group of $U^{\an}_{\bar \eta}$ in terms of $X_{\bar s}$ (\cite{mochi}).\\
 Let us make sure at first that we can get such a
description for the pro-$(p')$ completion, \ie the algebraic
fundamental group. One cannot apply directly Grothendieck's specialization
theorems (even if $U=X_{\eta}$) since the special fiber is not smooth but only
semistable. Indeed, a pro-$(p')$ geometric covering of the generic fiber will
generally only induce a Kummer covering on the special fiber. These are
naturally described in terms of \emph{log geometry}, more precisely in
terms of \emph{Kummer-étale} (két) coverings of a log scheme. One can endow $X$ (and thus
$X_s$ too by restriction)
with a natural log structure such that the pro-$(p')$ fundamental group of
$U$ is isomorphic to a pro-$(p')$ log fundamental group (as defined
in~\cite{ill}) of $X_{\bar s}$. One then gets a description of
$\ga(U_{\bar\eta})$ by taking the projective limit under tame coverings of
$K$, or equivalently under két coverings of $s$ endowed with its natural
log structure: there is an equivalence between finite étale coverings of $U_{\bar
  \eta}$ and ``geometric két coverings'' of $X_{\bar s}$.\\
A két covering of $X$ will still be a semistable model of its generic fiber
if one replaces $K$ by some tame extension. Thus, one can describe the
topology of the corresponding covering of $U_{\bar \eta}$ in terms of the
graph of the corresponding geometric két covering of $X_{\bar s}$.\\


Let us now come back to the problem of cospecialization. Let $X\to Y$ be a semistable
curve over $O_K$ with $X\to Y$ endowed with compatible log structure (see definition~\ref{stlogcurve}).\\
Let $\bar \eta_1$ (resp. $\bar \eta_2$) be a (Berkovich) geometric point of
$Y_0:=Y^{\an}_{\tr}\cap \fk Y_\eta\subset Y^{\an}_K$, where $Y_{\tr}$ is the locus of $Y$ where the log structure is trivial and $\fk Y_\eta$ is the generic fiber of the formal completion of $Y$ along its closed fiber (if $Y$ is proper, then $Y_0=Y^{\an}_{\tr}$). Let $\bar s_1$ (resp. $\bar s_2$) be its log reduction in $Y_s$ .\\
To use the previous description of the tempered fundamental group of
$U_{\bar \eta_1}$ and $U_{\bar \eta_2}$ in terms of $X_{\bar s_1}$ and
$X_{\bar s_2}$, we have to assume that $\bar \eta_1$ and $\bar \eta_2$ lie
over Berkovich points with discrete valuation.\\

The main result of this paper is the following:
\begin{thm}[{th. \ref{cospcourbes}}]\label{sp}Let $K$ be a complete discretely valued field. Let $\mbb L$ be a set of primes that does not contain the residual characteristic of $K$. Let $Y\to O_K$ be a morphism of log schemes. Let $Y_0=Y^{\an}_{\tr}\cap \fk Y_\eta\subset Y^{\an}$ where $\fk Y$ is the completion of $Y$ along its closed fiber. Let $X\to Y$ be a proper semistable
curve  with compatible log structure. Let $U=X_{\tr}$. Let $\eta_1$ and $\eta_2$ be two Berkovich points of $Y_0$ whose residue fields have discrete valuation, and let $\bar \eta_1,\bar \eta_2$ be
  geometric points above them. 
Let $\bar s_2\to\bar s_1$ be a log specialization
  of their log reductions such that there exists a compatible specialization $\bar \eta_2\to\bar \eta_1$, then there is a
  cospecialization homomorphism $\gtemp(U_{\bar\eta_1})^{\mbb L
    }\to\gtemp(U_{\bar\eta_2})^{\mbb L}$. Moreover, it is an isomorphism if $\overline M_{Y,\bar s_1}\to\overline M_{Y,\bar s_2}$ is an isomorphism.\end{thm}

Let us come back to our example of the moduli space of pointed stable elliptic curves with
high enough level
structure $M$ over $O_K$, and let $C$ be the canonical stable elliptic curve on
$M$. If $\eta_1$ and $\eta_2$ are two Berkovich points of $M_\eta$, they
are in $M_\eta^{\tr}$ if and only if $C_{\eta_1}$ and $\C_{\eta_2}$ are
smooth. $C\to M$, endowed with their natural log-structures over
$(O_K,O_K^*)$, is a semistable morphism of log schemes. One thus get a cospecialization
outer morphism $\gtemp(C_{\overline \eta_1})\to\gtemp(C_{\overline \eta_2})$ for
every specialization $\overline s_2\to\overline s_1$, which is an
isomorphism if $\overline s_1$ and $\overline s_2$ are in the same stratum
of $M_s$. Since the moduli stack
of pointed stable elliptic curves over $\Spec k$ has only two strata, one
corresponding to smooth elliptic curves $M_0$ and one to singular curves $M_1$,
one gets that $\gtemp(E_1)^{(p')}\simeq\gtemp(E_2)^{(p')}$ if $E_1$ and $E_2$
are two curves with good reduction or two Tate curves (the isomorphism
depends on choices of cospecializations). Since $M_1$ is in the closure of
$M_0$ one gets a morphism from the tempered fundamental group of a Tate
curve to the tempered fundamental group of an elliptic curve with good
reduction.\\

The first thing we need in order to construct the cospecialization homomorphism for
tempered fundamental groups is a
specialization morphism between the $(p')$-log geometric fundamental groups of
$X_{\bar s_1}$ and $X_{\bar s_2}$. Such a specialization morphism will be
constructed by proving that one can extend any $(p')$-log geometric
covering of $X_{s_1}$ to a két covering of $X_U$ where $U$ is some két
neighborhood of $s_1$. If one has
such a specialization morphism, by comparing it to the fundamental groups
of $X_{\bar\eta_1}$ and $X_{\bar\eta_2}$ and using Grothendieck's
specialization theorem, we will easily get that it must be an
isomorphism. This specialization morphism is easily deduced
from~\cite{org2} if $s_1$ is a strict point of $Y$ (\ie the log structure
of $s_1$ is simply the one induced by $Y$), \ie the log
structure of $s_1$ is just the pull back of the log structure of $Y$, but
is not straightforward when the log structure is really modified. Thus
we will study the invariance of the log geometric fundamental group by
change of fs base point. The main result we will prove (in any dimension) is the following~:
\begin{thm}[{th. \ref{chgmtbase1}}]\label{chbase} Let $s'\to s$ be a morphism of fs log points
  with isomorphic algebraically closed underlying fields. Let $X\to s$ be a
  saturated morphism of log schemes with $X$ noetherian and let $X'\to
  s'$ be the pull back to $s'$. Then the map
  $\ggeom(X'/s',\bar x')\to\ggeom(X/s,\bar x)$  is an isomorphism.\end{thm}
It is interesting to notice that, in this situation, this is an isomorphism
for the full fundamental group, and not only of the pro-$(p')$ part. This
mainly comes from the fact that the morphism of underlying schemes $\mring
X'\to\mring X$ is an isomorphism (so that the problem only comes from the
logarithmic structure and not the schematic structure). This result is
proved by a local study on $X$ for the strict étale topology. The log geometric
fundamental group can be described in terms of an inductive filtered limit of
categories of coverings. One then has to study an inductive filtered
limit of stacks over $X_{\et}$.\\

 Then we have to construct cospecialization topological morphisms for a
 semistable curve, more precisely cospecialization morphisms of the graphs
 of the geometric fibers. This will be done étale locally. These morphisms are not morphisms of graphs
 in the usual sense, since an edge can be contracted over a vertex, but
 still give a map between their geometric realizations, whence a 
 map of homotopy types $U_{\bar\eta_1}\to U_{\bar\eta_2}$. This can also be
 done for any két covering of $X_{\bar s_1}$: we thus get such a map of homotopy
 types for every $(p')$-covering of $U_{\bar\eta_1}$. Those maps are
 compatible, and thus glue together to give the wanted cospecialization of tempered
 fundamental groups.\\


\subsection*{}
The paper is organized as follows.\\

In the first section, we will recall the basic notions about tempered
fundamental groups and define the $(p')$-versions. We will also recall what
we will need about graphs.\\


In the second section, we will recall the definition of log
fundamental groups. We will then study specialization of log fundamental
groups in paragraph \ref{speclog}.\\

Finally, in a third part, we will construct the cospecialization morphisms
of graphs of the geometric fibers of a semistable curve. We will verify the
compatibility with két morphisms to obtain the wanted cospecialization
morphisms of tempered fundamental groups.

%\begin{acknowledgements}
This work is part of a PhD thesis. I would like to thank my advisor, Yves André, for
suggesting me to work on the cospecialization of the tempered fundamental
group and taking the time of reading and correcting this work. I would also
like to thank Luc Illusie and Fumiharu Kato for taking interest in my
problem about the invariance of geometric log fundamental groups by base change.
%\end{acknowledgements}

\section{Preliminaries}
\subsection{Tempered fundamental groups}
Let $K$ be a complete nonarchimedean field.\\
Let $\mbb L$ be a set of prime numbers (for example, we will denote by
$(p')$ the set of all primes except the residual characteristic $p$ of
$K$). An $\mbb L$-integer will be an integer which is a product of elements
of $\mbb L$.\\
If $X\subset \overline X$ is a $K$-algebraic variety, $X^{\an}$ will be the
$K$-analytic space in the sense of Berkovich associated to $X$.
A morphism $f:S'\to S$ of analytic spaces is said to be an \emph{étale covering} if $S$ is covered by open subsets $U$ such that $f^{-1}(U)=\coprod V_j$ and $V_j\to U$ is étale finite~(\cite{dJ1}).\\
For example, étale $\mbb L$-finite coverings (\emph{i.e.} finite étale
coverings that are dominated by a Galois covering $S''$ of $S$ such that
$\#\Gal(S''/S)$ is an $\mbb L$-integer), also called \emph{$\mbb
  L$-algebraic coverings}, and coverings in the usual topological sense for
the Berkovich topology, also called \emph{topological coverings}, are
étale coverings.\\
Then, André defines tempered coverings in \cite[def. 2.1.1]{andre1}. We generalize this definition to
$\mbb L$-tempered coverings as follows:
\begin{dfn} \label{def:rvt:temp}
An étale covering $S' \to S$ is \emph{$\mbb L$-tempered} if it is a
 quotient of the composition of a topological covering $T' \to T$ and of a
 $\mbb L$-finite étale covering
 $T \to S$.
\end{dfn}
This is equivalent to say that it becomes a topological covering after
pullback by some $\mbb L$-finite étale covering.\\
Let $X$ be a $K$-analytic space. We denote by $\Covtemp(X)^{\mbb L}$ (resp. $\Covalg(X)^{\mbb L}$,
$\Covtop(X)$) the category of $\mbb L$-tempered coverings (resp. $\mbb L$-algebraic coverings, topological coverings) of $X$ (with the obvious morphisms).\\

A geometric point of a $K$-analytic space $X$ is a morphism of Berkovich spaces $\mcal M(\Omega)\to X$ where $\Omega$ is an algebraically closed complete isometric extension of $K$.\\
Let $\bar x$ be a geometric point of $X$. Then one has a functor \[F^{\mbb L}_{\bar
x}:\Covtemp(X)^{\mbb L}\to\Set\] which maps a covering $S\to X$ to the set
$S_{\bar x}$.\\
The tempered fundamental group of $X$ pointed at $\bar x$ is
\[\gtemp(X,\bar x)^{\mbb L}=\Aut F^{\mbb L}_{\bar x}.\]
When $X$ is a smooth algebraic $K$-variety, $\Covtemp(X^{\an})^{\mbb L}$
and $\gtemp(X^{\an},\bar x)^{\mbb L}$ will also be denoted simply by
$\Covtemp(X)^{\mbb L}$ and $\gtemp(X,\bar x)^{\mbb L}$.\\
By considering the stabilizers $(\Stab_{F^{\mbb L}_{\bar x}(S)}(s))_{S\in \Covtemp(X)^{\mbb
L},s\in F^{\mbb L}_{\bar x}(S)}$ as a basis of open subgroups of $\gtemp(X,\bar
x)^{\mbb L}$, $\gtemp(X,\bar x)^{\mbb L}$ becomes a topological group. It is a prodiscrete topological group.\\
When $X$ is algebraic, $K$ of characteristic zero and has only countably
many finite extensions in a fixed algebraic closure $\overline K$,
$\gtemp(X,\bar x)^{\mbb L}$ has a countable fundamental system of
neighborhood of $1$ and all its discrete quotient groups are finitely
generated~(\cite[prop. III.2.1.7]{andre1}).\\
When $\mbb L$ is the set of all primes, we often forget it in the
notations. It should be remarked that in general, for a given $\mbb L$, one
cannot recover $\gtemp(X,\bar x)^{\mbb L}$ from $\gtemp(X,\bar x)$.\\

If $\bar x$ and $\bar x'$ are two geometric points, then $F^{\mbb L}_{\bar
x}$ and $F^{\mbb L}_{\bar x'}$ are (non canonically) isomorphic
(\cite[th. 2.9]{dJ1}). Thus, as usual, the tempered fundamental group
depends on the basepoint only up to inner automorphism (this topological
group, considered up to conjugation, will sometimes be denoted simply
$\gtemp(X)^{\mbb L}$).\\
The full subcategory of tempered coverings $S$ for which $F^{\mbb L}_{\bar
x}(S)$ is $\mbb L$-finite is equivalent to $\Covalg(S)^{\mbb L}$, hence
\[\big(\gtemp(X,\bar x)^{\mbb L}\big)^{\mbb L}=\ga(X,\bar x)^{\mbb L}\] (where
 $(\ )^{\mbb L}$ denotes the pro-$\mbb L$ completion).\\
For any morphism $X\to Y$, the pullback defines a functor
$\Covtemp(Y)^{\mbb L}\to\Covtemp(X)^{\mbb L}$. If $\bar x$ is a geometric
point of $X$ with image $\bar y$ in $Y$, this gives rise to a continuous
homomorphism \[\gtemp(X,\bar x)^{\mbb L}\to\gtemp(Y,\bar y)^{\mbb L}\]
(hence an outer morphism $\gtemp(X)^{\mbb L}\to\gtemp(Y)^{\mbb L}$).\\
One has the analog of the usual Galois correspondence:
\begin{thm}[({\cite[th. III.1.4.5]{andre1}})]  \label{galcorr} $F^{\mbb
L}_{\bar x}$ induces an equivalence of categories between the category of
direct sums of $\mbb L$-tempered coverings of $X$ and the category
$\gtemp(X,\bar x)^{\mbb L}\tSet$ of discrete sets endowed with a
continuous left action of $\gtemp(X,\bar x)^{\mbb L}$.\end{thm}

If $S$ is a $\mbb L$-finite Galois covering of $X$, its universal
topological covering $S^{\infty}$ is still Galois and every connected $\mbb
L$-tempered covering is dominated by such a Galois $\mbb L$-tempered covering.\\
If $((S_i,\bar s_i))_{i\in \mbf N}$ is a cofinal projective system (with
morphisms $f_{ij}:S_i\to S_j$ which maps $s_i$ to $s_j$ for $i\geq j$) of
geometrically pointed Galois $\mbb L$-finite étale coverings of $(X,\bar
x)$, let $((S^{\infty}_i,\bar s^{\infty}_i))_{i\in\mbf N}$ be the
projective system of its
pointed universal topological coverings (the transition maps will be
denoted by $f^{\infty}_{ij}$). It induces a projective system
$(\Gal(S_i^\infty/X))_{i\in\mbf N}$ of discrete groups. For every $i$,
$\Gal(S_i^\infty/X)$ can be identified with $F^{\mbb L}_{\bar
  x}(S_i^{\infty})$: this gives us compatible morphisms $\gtemp(X,\bar
x)^{\mbb L}\to\Gal(S_i^\infty/X)$.
Then, thanks to~\cite[lem. III.2.1.5]{andre1},
\begin{prop}\label{limproj} \[\gtemp(X,\bar x)^{\mbb L}\to\varprojlim \Gal(S^{\infty}_i/X)\] is an isomorphism.\end{prop}

In a more categorical way, we have a fibered category
$\Dtop(X)\to\Covalg(X)$, where the fiber $\Dtop(X)_S$ in an algebraic
covering $S$ of $X$ is $\Covtop(X)$.\\
Since algebraic coverings are of effective descent for tempered coverings,
the full subcategory of tempered coverings $T$ of $X$ such that $T_S\to S$
is a topological covering is naturally equivalent to the category
$\DDtemp_S$ of
descent data in the fibered category $\Dtop(X)$ with respect to $S\to
X$.\\
If $``\varprojlim"\ S_i$ is a universal procovering of $(X,x)$, one gets a
natural equivalence \[\Covtemp(X)=\projLim_i \DDtemp_{S_i}\]
In particular one can recover the tempered fundamental group from the
fibered category $\Dtop(X)\to\Covalg(X)$.\\
If $S\to S$ is an isomorphism, the induced functor
$\DDtemp_S\to\DDtemp_S$ is naturally isomorphic to the identity. Thus if
$\alpha: ``\varprojlim_i"\ S_i\to ``\varprojlim_i{}"\ S_i$ is an automorphism of
the universal pro-covering, the induced functor $\projLim_i
\DDtemp_{S_i}\to\projLim_i\DDtemp_{S_i}$ is naturally isomorphic to the
identity. Thus the construction does not depend of the choice of the
universal procovering.\\

To give a more stacky and functorial description, let us consider $\Covalg(X)$ with its
canonical topology. $\Dtop(X)$ is a separated prestack on $\Covalg(X)$ (\ie
just a fibered category; it is not a prestack of groupoids).\\
Let $\Dtemp(X)\to\Covalg(X)$ be the fibered category whose fiber over $U$ is
the category $\Covtemp(U)$ of tempered coverings of $U$. Then $\Dtemp(X)$
is a stack. The fully faithful cartesian functor of prestacks $\Dtop(X)\to\Dtemp(X)$ induces a
fully faithful cartesian functor of stacks $\Dtop(X)^a\to\Dtemp(X)$ where $\Dtop(X)^a$ is the stack
associated to $\Dtop(X)$. Since a tempered covering is a topological covering locally on
$\Covalg(X)$, this functor is in fact an equivalence~(\cite[th. II.2.1.3]{giraud}).\\
In a similar way:
\begin{prop}\label{stackydfn} The stack $(\Dtop(X)|_{\Covalg(X)^{\mbb
    L}})^a$ is the stack $\Dtemp(X)^{\mbb L}$ of $\mbb L$-tempered
coverings on $\Covalg(X)^{\mbb L}$.\end{prop}

\subsection{Graphs}
A \emph{graph} $\mbb G$ is given by a set of edges $\mcal E$ a set of vertices
$\mcal V$ and for any $e\in \mcal E$ a set of branches $\mcal B_e$ of
cardinality 2 and a map $\psi_e:\mcal B_e\to\mcal V$. A branch $b$ of $e$ can be
thought of as an orientation of $e$ (or a half-edge), and $\psi_e(b)$ is to be thought of as
the ending of $e$ when $e$ is oriented  according to $b$.\\
One can also equivalently replace the data of edges and branches of each
edge by the datum of the set of all branches $\mcal B=\coprod_e\mcal B_e$, with an involution
$\iota$ without fixed points, and a map $\psi:\mcal B\to \mcal V$. $\mcal E$ is
then the set of orbits of branches for $\iota$.\\
A \emph{genuine morphism of graphs} $\phi:\mbb G\to \mbb G'$ is given by a map
$\phi_{\mcal E}:\mcal
E\to \mcal E'$, a map $\phi_{\mcal V}:\mcal V\to\mcal V'$ and for every
$e\in \mcal E$ a bijection $\phi_e:\mcal B_e\to\mcal B'_{\phi_{\mcal
    E}(e)}$ such that the following diagram commutes:
\[\xymatrix{\mcal B_e \ar[d] \ar[r] & \mcal B'_{\phi_{\mcal E}(e)} \ar[d]\\
  \mcal V \ar[r] & \mcal V'}\]
Remark that $\phi_{\mcal E}$ and $\phi_{\mcal V}$ are not enough to define $\phi$ if $\mbb G$ has a loop (\ie an edge whose to branches abut to the same vertex): one can define an automorphism of $\mbb G$ just by inverting the two branches of the loop.\\

The topological cospecialization for semistable curves will be
given by maps of graphs which are not genuine morphisms.
A \emph{generalized morphism of graphs} $\phi:\mbb G\to\mbb G'$ will be given by:
\begin{itemize}
\item a map $\phi_{\mcal V}:\mcal V\to\mcal V'$,
\item a map $\phi_{\mcal E}:\mcal E\to\mcal E'\coprod\mcal V'$,
\item for any $e\in \mcal E$ such that $\phi_{\mcal E}(e)\in\mcal
  E'$, a bijection $\phi_e:\mcal B_e\to\mcal B'_{\phi_{\mcal
    E}(e)}$ such that the obvious diagram commutes (it is the same diagram
as in the case of genuine morphisms).\end{itemize} 
One can replace the last two data by the data of $\phi_{\mcal B}:\mcal
B\to\mcal B'\coprod\mcal V'$ such that if $\phi_{\mcal B}(b)\in\mcal B'$
then $\phi_{\mcal B}(\iota(b))=\iota'(\phi_{\mcal B}(b))$ and if $\phi_{\mcal
B}(b)\in\mcal V'$, $\phi_{\mcal B}(\iota(b))=\phi_{\mcal B}(b)$.\\
In particular, a genuine morphism is a generalized morphism. There is an
obvious composition of morphisms and generalized morphisms of graphs.\\
One thus gets a category $\Graph$ of graphs with genuine morphisms and a
category $\GenGraph$ of graphs with generalized morphisms.\\

There is a geometric realization functor $|\ |:\GenGraph\to\Top$
which maps a graph $\mbb G$ to \[|\mbb G|:=\Coker(\coprod_{b\in\mcal B} \pt_{1,b}\amalg\pt_{2,b}
\rightrightarrows\coprod_{v\in\mcal V} \pt_v\amalg\coprod_{b\in\mcal B}[1/2,1]_b),\]
where \begin{itemize}\item the upper map sends: 
\begin{itemize}\item $\pt_{1,b}$ to $1/2$ in $[1/2,1]_b$ \item
$\pt_{2,b}$ to $1$ in $[1/2,1]_b$,\end{itemize} \item the lower map sends \begin{itemize}\item$\pt_{1,b}$ to
$1/2$ in $[1/2,1]_{\iota(b)}$\item $\pt_{2,b}$ to $\pt_{\psi(b)}$.\end{itemize}\end{itemize}
If $\phi:\mbb G\to\mbb G'$ is a generalized morphism, $|\phi|$ is obtained
by mapping \begin{itemize}\item$\pt_v$ to $\pt_{\phi_{\mcal V}(v)}$,\item $[1/2,1]_b$ to
$[1/2,1]_{\phi_{\mcal B}(b)}$ if $\phi_{\mcal B}(b)\in \mcal B'$ (by the
identity of $[1/2,1]$),\item
$[1/2,1]_b$ to
$\pt_{\phi_{\mcal B}(b)}$ if $\phi_{\mcal B}(b)\in \mcal V'$.\end{itemize}
Remark that, if $\mbb G$ is just a loop, then the geometric realization of the morphism induced by inverting the two branches is not homotopic to the identity: thus $\phi_{\mcal E}$ and $\phi_{\mcal V}$ are not enough in general to characterize the topological behavior of $\phi$. 
\section{Specialisation of log fundamental groups}\label{loggeometry}
The main result of this part will be the invariance of the log geometric
fundamental group announced in theorem~\ref{chbase}. We will deduce from it
morphisms of specialization for the pro-$(p')$ log geometric fundamental
group of the fibers of a proper log smooth saturated morphism. 

\subsection{Log fundamental groups}
For a curve with bad reduction, one cannot apply Grothendieck's
specialization theorem to describe the geometric fundamental group of the
curve in terms of the fundamental group of its stable reduction since the
family is not smooth. However, such a comparison result exists in the realm
of log geometry. More precisely, if one considers a smooth and proper variety with
semistable reduction, the semistable model can naturally be endowed with a
log structure, and the pro-$(p')$ fundamental group of the variety is
canonically isomorphic to the pro-$(p')$ log fundamental group of the
semistable reduction. Here we recall the basic definitions and results
about log fundamental groups.\\

First, recall some usual notations about log schemes. If $X$ is a log
scheme, the sheaf of monoid defining its log structure will usually be
denoted by $M_X$, the underlying scheme will be denoted by $\mring{X}$, and
the open subset of $\mring X$ where the log structure is trivial will be
denoted $X_{\tr}$.\\
If $P$ is a monoid, one denotes by $\Spec P$ the set of primes of $P$. There is a natural map $\Spec \mbf Z[P]\to\Spec P$.

\begin{dfn}A morphism $h:Q\to P$ of fs monoids is \emph{Kummer} (resp. $\mbb
L$-Kummer)
if $h$ is injective
and for every $a\in P$, there exists a positive integer (an $\mbb L$-integer) $n$ such that $a^n\in h(Q)$
(note that if $Q\to P$ is Kummer, $\Spec P\to\Spec Q$ is an homeomorphism).\\
A morphism $f:X\to Y$ of fs log schemes is said to be \emph{Kummer} (resp. \emph{exact})
if for
every geometric point $\bar x$ of $X$, $\overline M_{Y,f(\bar x)}\to \overline
M_{X,\bar x}$ is Kummer (resp. exact).\\
A morphism of fs log scheme is \emph{Kummer étale} (or két for short) if it
is Kummer and log étale.\end{dfn}
A morphism $f$ is két if and only if étale locally it is deduced by strict
base change and étale localization from a map $\Spec \mbf Z[P]\to\Spec \mbf
Z[Q]$ induced by a Kummer map $Q\to P$ such that $nP\subset Q$ for some $n$
invertible on $X$.\\
In fact if $f:Y\to X$ is két, $\bar y$ is a geometric point of $Y$, and
$P\to M_X$ is an exact chart of $X$ at $f(\bar y)$, there is an étale
neighborhood $U$ of $\bar x$ and a Zariski open neighborhood $V\subset
f^{-1}(U)$ of $\bar y$ such that $V\to U$ is isomorphic to $U\times_{\Spec
  \mbf Z[P]}\Spec\mbf Z[Q]$ with $P\to Q$ a $\mbb L$-Kummer morphism where
$\mbb L$ is the set of primes invertible on $U$ (\cite[Prop. 3.1.4]{stix}).\\
Két morphisms are open and quasi-finite.\\

The category of két fs log schemes over $X$ (any $X$-morphism
between two such fs log schemes is then két) where the covering families
$(T_i\to T)$ of $T$ are the families that are set-theoretical covering
families (being a set-theoretical covering két family is stable under fs base change) is a site. We
will denote by $X_{\ket}$ the corresponding topos.\\
Any locally constant finite object of $X_{\ket}$ is representable. A két fs
log scheme over $X$ that represents such a locally constant finite sheaf
will be called a \emph{két covering} of $X$. We will
denote by $\KCov(X)$ the category of két coverings
of $X_{\ket}$.\\
A log geometric point is a log scheme $s$ such that $\mring{s}$ is the
spectrum of a
separably closed field $k$ and $M_s$ is saturated and multiplication
by $n$ on $\overline
M_s$ is an isomorphism for every $n$ prime to the characteristic of $k$.\\
A log geometric point of $X$ is a morphism $x:s\to X$ of log schemes where $s$ is a log
geometric point. A két neighborhood $U$ of $s$ in $X$ is a morphism $s\to
U$ of $X$-log schemes where $U\to X$ is két. Then if $x$ is a log geometric
point of $X$, the functor $F_x$ from $X_{\ket}$ to $\Ens$ defined by $\mcal F\mapsto
\varinjlim_U \mcal F(U)$ where $U$ runs through the directed category of két
neighborhoods of $x$ is a point of the topos $X_{\ket}$ and any point of
this topos is isomorphic to $F_x$ for some log geometric point and the family of points $(F_x)$ where $x$ runs through log geometric points of $X$
is a conservative system of points.\\
 One also defines the log strict
  localization $X(x)$ to be the inverse limit in the category of saturated
log schemes of the két neighborhoods of $x$. If $x$ and $y$ are log
geometric points of $x$, a \emph{specialization} of log geometric points $x\to y$
is a morphism $X(x)\to X(y)$ over $X$.\\
A specialization $x\to y$ induces a canonical morphism  $F_y\to F_x$ of functors.\\
 If there is a specialization $x\to
y$ of the underlying topological points, then there is some specialization $x\to y$ of log
geometric points.\\
If $X$ is connected, for any log geometric point $x$ of $X$, $F_x$ induces
a fundamental functor $\KCov(X)\to\fSet$ of the Galois category
$\KCov(X)$.\\
One then denotes by $\glog(X,x)$ the profinite group of automorphisms of
this fundamental functor.\\

Strict étale surjective morphisms satisfy effective descent for két
coverings (\cite[prop. 3.2.19]{stix}).\\
If $f:S'\to S$ is an exact morphism of fs log schemes such that $\mring{f}$
is proper, surjective and of finite presentation, then $f$ satisfies
effective descent for két coverings (\cite[th. 3.2.25]{stix}).\\


Let us now state the main results to compare log fundamental groups of
different log schemes (in particular specialization comparisons). According to \cite[th. 7.6]{ill}, if $X$ is a log regular fs log scheme, $\KCov(X)$ is equivalent to the category of tamely ramified coverings of $X_{\tr}$. If $\mbb L$ is a set of primes invertible on $X$, by taking the pro-$\mbb L$ completion, one gets:
\begin{thm} If $X$ is a log regular fs log scheme and all the primes of $\mbb L$ are
invertible on $X$, then $\KCov(X)^{\mbb L}\to\Covalg(X_{\tr})^{\mbb L}$ is
an equivalence of categories.\end{thm}
For example, if $X$ is a regular scheme and $D$ is a normal crossing divisor, $M_X=O_X\cap j_*O^*_{X\backslash D}$ is a log structure on $X$ for which $X$ is log regular and $X_{\tr}=U:=X\backslash D$. Thus there is an equivalence of categories $\KCov(X)^{\mbb L}\to\Covalg(U)^{\mbb L}$.
\begin{prop}[({\cite[cor. 2.3]{org2}})] \label{orgsp} Let $S$ be a strictly local scheme
  with closed point $s$ and let $X$ be a connected fs log scheme such that
  $\mring X$ is proper over $S$. Then
\[\KCov(X)\to\KCov(X_s)\]is an equivalence of categories.\end{prop}
One can extend this result to henselian schemes:
\begin{thm}\label{henselspec}
Let $S$ be a henselian scheme
  with closed point $s$, and let $X$ be a connected fs log scheme such that
  $\mring X$ is proper over $S$. Then
\[\KCov(X)\to\KCov(X_s)\] is an equivalence of categories.
\end{thm}
\dem
First assume $X_s$ to be geometrically connected. Then $X$ is also connected and we have to prove that $\glog(X_s)\to\glog(X)$. Let $\overline s$ be a strict localization of $s$ and let $\overline S$ be the strict localization of $S$ at $\overline s$. Let $S_i$ be a pointed Galois covering of $S$, let $G_i$ be its Galois group and let $s_i=s\times_SS_i$. Then we have a diagram with exact lines:
\[\xymatrix{1 \ar[r] & \glog(X_{s_i}) \ar[r]\ar[d] & \glog(X_s) \ar[d]\ar[r] & G_i \ar[r]\ar[d] & 1\\
1 \ar[r] & \glog(X_{S_i}) \ar[r] & \glog(X) \ar[r] & G_i \ar[r] & 1
}
\]
By taking the projective limit when $S_i$ describes the category of pointed Galois covering of $S$, one gets a diagram with exact lines
\[\xymatrix{1 \ar[r] & \varprojlim_{S_i}\glog(X_{s_i}) \ar[r]\ar[d] & \glog(X_s) \ar[d]\ar[r] & \ga(S) \ar[r]\ar[d] & 1\\
1 \ar[r] & \varprojlim_{S_i}\glog(X_{S_i}) \ar[r] & \glog(X) \ar[r] & \ga(S) \ar[r] & 1
}
\]
But $\glog(X_{\overline S})\to \varprojlim_{S_i}\glog(X_{S_i})$ is an isomorphism (for this one does not need $X\to S$ to be proper but simply to be quasicompact and separated). To show this is equivalent to prove that $F:\injLim\KCov(X_{S_i})\to \KCov(X_{\overline S})$ is an equivalence of category. If $T_i\to X_{S_i}$ is a k\'et covering such that $T_i\times_{X_{S_i}}X_{S_j}$ is connected for every $j$, then $T_i\times_{X_{\overline S}}$ is also connected. Thus $\injLim\KCov(X_{S_i})\to \KCov(X_{\overline S})$ is fully faithful. Let $T\to X_{\overline S}$ be a covering. Let $\coprod_{k\in K} U_k\to X_{overline S}$ be a surjective etale morphism such that $U_k$ has a chart $U_k\to\Spec \mbf Z[P]$, and there is an \'etale morphism $T_k:=T\times_{X_{\overline S}} U_k\to U_{k,Q}:=U_k\times_{\Spec\mbf Z[P]}\Spec\mbf Z[Q]$. Assume $K$ to be finite and $U_k$ to be quasicompact. Then there exists $i$ and $U_{k,i}\to X_{S_i}$ such that $U_k=U_{k,i}\times_{X_{S,i}}X_{\overline S}$. If $i$ is big enough, there is a chart $U_{k,i}\to\Spec \mbf Z[P]$ extending the chart of $U_k$ and $U_{k,Q}=U_{k,Q,i}\times_{X_{S_i}}X_{\overline S}$ where $U_{k,Q,i}=U_{k,i}\times_{\Spec \mbf Z[P]}\Spec \mbf Z[Q]$. If $i$ is big enough, the \'etale morphism $T_k\to U_{k,Q}$ comes from some \'etale morphism $T_{k,i}\to U_{k,Q,i}$.  As $F$ is fully faithful for any $X$ quasicompact and separated over $S$, for $i$ big enough, $(T_{k,i})_k$ can be enriched into a descent datum with respect to $\coprod_k U_{k,i}\to X_{S_i}$ inducing after pullback to $X_{\overline S}$ the descent datum given by $T$. Since surjective \'etale morphisms satisfy effective descent for k\'et coverings, one gets for $i$ big enough a k\'et covering $T_i\to X_{S_i}$ such that $T=T_i\times_{X_{S_i}}X_{\overline S}$. Thus $\glog(X_{\overline S})\to \varprojlim_{S_i}\glog(X_{S_i})$ is an isomorphism.\\
Similarly $\glog(X_{\overline s})\to \varprojlim_{S_i}\glog(X_{s_i})$ is an isomorphism. Thanks to proposition \ref{orgsp}, $\glog(X_{\bar s})\to\glog(X_{\overline S})$ is an isomorphism. Thus $\glog(X_s)\to\glog(X)$ is also an isomorphism.\\
In the general case, let $X\to S'$ be the Stein factorization of $X\to S$. For every connected component $S'_j$ of $S'$, let $X_j=X\times_{S'}S'_j$. Since $S'_j$ is henselian and $X_j\to S'_j$ has geometrically connected fibers, one gets that $\KCov(X_{j,s})\to\KCov(X_j)$ is an equivalence of category. Since $\KCov(X)=\prod_j\KCov(X_j)$ and $\KCov(X_s)=\prod\KCov(X_{j,s})$, one gets that $\KCov(X)\to\KCov(X_s)$ is an equivalence of categories.
\findem
\begin{cor}\label{speclogdvr} Let $O_K$ be a complete discretely valued ring endowed with its natural log
structure and let $\mbb L$ a set of prime numbers invertible in $O_K$. Let  $X\to \Spec O_K$ be a proper and log smooth morphism and
let $U:=X_{\tr}\subset X_{\eta}$. There is a natural
equivalence of categories \[\KCov(X_s)^{\mbb L}\simeq\Covalg(U)^{\mbb
  L}.\]\end{cor}
In particular, if $\mring X\to\Spec O_K$ is a semistable model of
$X_{\eta}$, and the log structure on $X$ is given by $M_X=O_X\cap j_*O^*_{X_{\eta}}$ where $j:X_{\eta}\to X$, then $X\to\Spec O_K$ is log smooth and
$X_{\tr}=X_\eta$. We get an equivalence of categories  $\KCov(X_s)^{\mbb L}\simeq\Covalg(X_{\eta})^{\mbb
  L}$, and thus an isomorphism
\[\ga(X_{\eta})^{\mbb L}\to\glog(X_s)^{\mbb L}.\]

Here we recall basic results about saturated morphisms of fs log
schemes. The main reference on the subject is~\cite{sattsuji}, which is
unfortunately unpublished.\\
\begin{dfn}A morphism of fs monoids $P\to Q$ is \emph{integral} if, for any morphism
of integral monoids $P\to Q'$, the amalgamated sum $Q\oplus_PQ'$ is still
integral.\\
A integral morphism of fs monoids $P\to Q$ is \emph{saturated} if, for any morphism
of fs monoids $P\to Q'$, the amalgamated sum $Q\oplus_PQ'$ is still
fs.\\
A morphism $f:Y\to X$ of fs log schemes is \emph{saturated} if for any geometric
point $\bar y$ of $Y$, $\bar M_{X,f(\bar y)}\to\bar M_{Y,\bar y}$ is
saturated.\end{dfn}
If $Y\to X$ is saturated and $Z\to X$ is a morphism of fs log schemes, then
the underlying scheme of $Z\times_XY$ is $\mring Z\times_{\mring X}\mring
Y$.\\ 
If $P\to Q$ is a local and integral (resp. saturated) morphism of fs
monoids and $P$ is sharp, the morphism $\Spec \mbf
Z[Q]\to\Spec\mbf Z[P]$ is flat (resp. separable, \emph{i.e.} flat
with geometrically
reduced fibers, cf.~\cite[cor. 4.3.16]{ogus} and \cite[rem. 6.3.3]{satmorph}).\\
Let $f:X\to Y$ be log smooth, let $\bar x$ be a geometric point of
$X$ and let $\bar y$ be its image in $Y$. Etale locally on $Y$, there is a chart 
$Y\to\Spec P$ such that $P\to\overline M_{Y,\bar y}$ is an isomorphism. Then, according to~\cite[th. 3.5]{kato},
there is étale locally at $x$ a fs chart $\phi:P\to Q$ of $Y\to X$ such that $Y\to
\Spec \mbf Z[Q]\times_{\mbf Z[P]}X$ is étale such that $\phi$ is injective and the torsion part of $\Coker(\phi^{\gp})$ has order invertible on $X$. Up to localizing $Q$ by the face corresponding to $\bar x$, one can assume that $X\to\Spec Q$ is exact at
$x$. Thus if $f$ is integral (resp. saturated), $P\to Q$ is a local and
integral (resp. saturated) morphism of fs monoids and $P$ is sharp. Thus
$f$ is flat (resp. separable).\\ 
If $P\to Q$ is an integral morphism of fs monoids, there exists an integer
$n$ such that the pullback $P_n\to Q'$ of $P\to Q$ along
$P\stackrel{n}{\to}P=P_n$ is saturated (theorem~\cite[A.4.2]{satmorph}).\\
Moreover if $P\to Q$ factors through $Q_0$ such that $P\to Q_0$ is saturated
and $Q_0\to Q$ is $\mbb L$-Kummer, $n$ can be chosen to be an $\mbb
L$-integer.\\

\subsection{Log geometric fundamental groups}

Let $X\to s$ be a morphism of fs log schemes, where $s$ is an fs log point. Let
$\bar x$ be a log geometric point of $X$ and let $\bar s$ be its image in
$s$.\\
Then we define the \emph{log geometric fundamental group} of $X$ at $\bar x$ to be
\[\ggeom(X/s,\bar x)=\Ker(\glog(X,\bar x)\to\glog(s,\bar s)).\]
If $(t,\bar t)\to (s,\bar s)$ is a Galois connected pointed két covering of $(s,\bar
s)$, one defines $X_t=X\times_st$ and $\bar x_t=(\bar x,\bar t)$.\\
Then one has an exact sequence:
\[1\to \glog(X_t,\bar x_t)\to\glog(X,\bar x)\to\Gal(t/s),\]
and the right map is onto if $X_t$ is connected (we will say that $X$ is
\emph{log geometrically connected} if $X_t$ is connected for any connected
két covering $t$ of $s$).\\
By taking the projective limit of the previous exact sequence when $(t,\bar
t)$ runs through the directed category of pointed Galois connected
coverings of $(s,\bar s)$, one gets a canonical isomorphism $\ggeom(X/s,\bar
x)=\varprojlim_{(t,\bar t)} \glog(X_t,\bar x_t)$.\\
Actually, if $\tilde t\to t$ is the reduced subscheme of $t$ endowed with the
inverse image log structure, the map $\glog(X_{\tilde t},\bar
x_t)\to\glog(X_t,\bar x_t)$ is an isomorphism, so that one can replace $X_t$
in the previous projective limit by $X_{\tilde t}$.\\
The category $\ggeom(X/s,\bar x)\tfEns$ is naturally equivalent to the
category \[\KCovgeom(X/s):=\projLim_{\tilde t} \KCov(X_{\tilde t}).\]

If one has a commutative square of pointed fs log schemes:
\[\xymatrix{(X',\bar x') \ar[r]^{\phi} \ar[d] & (X,\bar x) \ar[d]\\
(s',\bar s')\ar[r]^{\psi} & (s,\bar s)}\]
where $s'$ and $s$ are log points, one gets a commutative diagram of
profinite groups:
\[\xymatrix{\glog(X',\bar x') \ar[r] \ar[d] & \glog(X,\bar x) \ar[d]\\
\glog(s',\bar s')\ar[r] & \glog(s,\bar s)}\]
By taking the kernel of the vertical arrows, one gets a map
$\ggeom(X'/s',\bar x')\to\ggeom(X/s,\bar x)$, which is functorial with respect
to $(\phi,\psi)$.\\
One also has, by definition of the universal pro-pointed covering of
$(s',\bar s')$, a canonical morphism of pro-$(s',\bar s')$-pointed két
covering
\[\varprojlim_{(t',\bar t')} (t',\bar t')\to \varprojlim_{(t,\bar t)}
\psi^*(t,\bar t)\](where $(t',\bar t')$ runs through pointed
Galois connected két
coverings of $(s',\bar s')$ and $(t,\bar t)$ runs through pointed Galois
connected két coverings of $(s,\bar s)$), and thus a
morphism of pro-pointed fs log schemes \[\varprojlim_{(t',\bar t')} (t',\bar
t')\to \varprojlim_{(t,\bar t)} (t,\bar t).\]
This induces a morphism of pro-pointed fs log schemes
\[\varprojlim (X'_{t'},\bar x'_{t'})\to \varprojlim (X_t,\bar x_t),\]
hence a
map of profinite groups \[\varprojlim \glog(X'_{t'},\bar
x'_{t'})\to\varprojlim \glog(X_t,\bar x_t),\]
such that the following square of profinite groups is commutative:
\[\xymatrix{\varprojlim\glog(X'_{t'},\bar x'_{t'})\ar[r]\ar@{=}[d] & \varprojlim\glog(X_t,\bar
x_t)\ar@{=}[d]\\ \ggeom(X'/s',\bar x')\ar[r] & \ggeom(X/s,\bar x)}\]

Let $O_K$ be a complete discretely valued ring endowed with its natural log
structure and let $\mbb L$ a set of prime numbers invertible in $O_K$. Let  $X\to \Spec O_K$ be a proper and log smooth morphism and
let $U:=X_{\tr}\subset X_{\eta}$. There
is a geometric analog to the specialization isomorphism $\ga(X_{\eta})^{\mbb L}\to\glog(X_s)^{\mbb L}$ of corollary \ref{speclogdvr}:
\begin{thm}[{\cite[th. 1.4]{kisin}}]\label{speclogdvrgeom} There is a natural equivalence of categories
  \[\KCovgeom(X/s)^{\mbb L}\simeq\Covalg(U_{\bar\eta})^{\mbb L}.\]
\end{thm}
It can be deduced from corollary \ref{speclogdvr} thanks to the fact that
 any algebraic covering of $U_{\bar
  \eta}$ is already defined over a tamely ramified extension of $K$ (\cite[prop. 1.15]{kisin}).
\subsection{Semistable log curves}
\begin{dfn}\label{stlogcurve}
A morphism $X\to S$ of fs log schemes is a \emph{semistable log curve} if étale
locally on $S$ there is a chart $S\to \Spec P$ such that one of the
following is satisfied:
\begin{itemize}
\item $X\to S$ is a strict smooth curve,
\item $X\to S$ factorizes through a strict étale morphism $X\to S\times_{\Spec\mbf Z[P]}\Spec\mbf
    Z[Q]$ with $Q=(P\oplus<u,v>)/u + v=p$ and $p\in P$,
\item $X\to S$ factorizes through a strict étale morphism $X\to S\times_{\Spec\mbf Z[P]}\Spec\mbf
    Z[P\oplus\mbf N]$.
\end{itemize}
\end{dfn}
A semistable log curve is \emph{strictly semistable} if étale locally on $S$, there
are such maps locally for the Zariski topology of $X$.\\

\begin{prop}
A morphism $X\to S$ is a semistable log curve if and only if it is a log smooth and
saturated morphism purely of relative dimension 1.\\
\end{prop}
\dem
The direct sense is obvious. Let $X\to S$ be a saturated log smooth scheme of pure dimension 1. As the definition of a semistable log curve is local on the \'etale topology of $X$ and $S$, one can assume that $S$ has a chart $S\to\Spec\mbf Z[P]$ and $X=S\times_{\Spec\mbf Z[P]}\Spec\mbf Z[Q]$ where $P\to Q$ is an injective local and saturated morphism of monoids, $P$ is sharp and $Q^{\gp}/P^{\gp}$ is invertible on $S$. In particular $T^{\gp}:=\overline Q^{\gp}/\overline P^{\gp}$ is torsionfree. Since $P\to Q$ is saturated, $\Spec\mbf Z[P]\to\Spec\mbf Z[Q]$ is flat and $1=\dim \Spec\mbf Z[P]-\dim \Spec\mbf Z[Q]=\rk P^{\gp}-\rk Q^{\gp}=\rk Q^{\gp}/P^{\gp}\geq \rk T^{\gp}$. Thus $T^{\gp}$ is $\{1\}$ or $\mbf Z$. . For every $x\in T$, there exists a unique  $\psi(x)\in \overline Q$ such that $f^{-1}(x)\cup Q=\psi(x)+\overline P$ where $f:\overline Q^{\gp}\to T^{\gp}$ (\cite[prop. I.4.3.14]{ogus}). In particular, if $T^{\gp}=\{1\}$, then $\overline P\to\overline Q$ is bijective, thus $X\to S$ is strict and thus $X\to S$ is smooth.\\
Assume $T^{\gp }=\mbf Z$. Then $\rk Q^{\gp}/P^{\gp}=\rk T^{\gp}$ and thus $\rk Q^{\gp}=\rk \overline Q^{\gp}$. Since $\overline Q^{\gp}$ is a free abelian group, one can choose a splitting $Q=\overline Q\oplus Q^*$. Since $Q^*\hookrightarrow Q^{\gp}/P^{\gp}$ is finite of order invertible on $S$, $X\to S\times_{\Spec \mbf Z[P]}\Spec\mbf Z[\overline Q]$ is \'etale. Thus one can assume that $Q$ is sharp. Let $T$ be the image of $Q$ in $T^{\gp}$. Then $T=\mbf N$ or $T=\mbf Z$.\\
First assume $T=\mbf N$. Then $n\psi(1)=\psi(n)+p$ with $p\in P$. Since $P\to Q$ is saturated and $p\leq n\psi(1)$, there exists $p'\in P$ such that $p\leq np'$ and $p'\leq \psi(1)$. Thus, by definition of $\psi(1)$, $p'=0$, thus $p=0$ and $\psi(n)=n\psi(1)$. Thus $Q=P\oplus \mbf N\psi(1)$.\\
If $T=\mbf Z$, let $u=\psi(1)$ and $v=\psi(-1)$. Since $\psi(u+v)=0$, $p:=u+v\in P$. As in the previous case, if $n\geq 0$, then $\psi(n)=n\psi(1)$ and $\psi(-n)=n\psi(-1)$. Thus $Q=P\oplus<u,v>/(u+v=p)$.
\findem
The underlying morphism of schemes $\mring X\to \mring S$ is a semistable
curve of schemes. In particular, if $\mring S$ is a geometric point, one can associate to $X$ a graph $\mbb G(X)$ in the
following way: the vertices are the irreducible components of $X$,
the edges are the nodes. If $x$ is a node, then the henselization $X(x)$ of $X$ at $x$ has two irreducible components: these components are the branches of the edge corresponding to $x$. If $z$ is an irreducible component of $X(x)$ and $z'$ is the irreducible component containing the image of $z$ in $X$, the branch corresponding to $z$ abuts to the vertex corresponding to $z'$ (this graph does not depend of the
log structure).\\

If $X\to S$ is a proper semistable log curve and $X'\to X$ is a két
covering, then for any log geometric point $\bar s$ of $S$, there is a két
neighborhood $U$ of $\bar s$ such that $X'_U\to U$ is saturated. Then
$X'_U\to U$ is also a semistable curve.\\
The morphism $\mring X'_{\bar s}\to\mring X_{\bar s}$ induces a
genuine morphism $\mbb G(X'_{\bar s})\to\mbb G(X_{\bar s})$ of graphs. 


If $K$ is a complete nonarchimedean field with separably closed residue field $k$, $O_K$ is its ring of integers and $X\to
O_K$ is a proper semistable curve with smooth generic fiber, there is a canonical embedding $|\mbb G(X_k)|\to
X^{\an}_\eta$ which is a homotopy equivalence (\cite[th. 4.3.2]{berk}). It is compatible with any
isometric extension of $K$.\\
Moreover, if $U$ is any dense Zariski open subset of $X_{\eta}$, $|\mbb
G(X_k)|$ is mapped into $U^{\an}$ and $|\mbb G(X_k)|\to U^{\an}$ is still a
homotopy equivalence.\\
If $X\to O_K$ is a semistable log curve and $X'\to X$ is a két morphism
such that $X'$ is still a semistable curve, the following diagram is commutative:
\[\begin{array}{ccc} |\mbb G(X'_k)| & \to & {X'}_{\eta}^{\an}\\ \dar & & \dar\\
|\mbb G(X_k)| & \to & X_{\eta}^{\an}\end{array}\]







\subsection{Specialization of log fundamental groups}\label{speclog}

Let us study specialization of log geometric fundamental groups (that is
the projective limit of the log fundamental groups after taking két
extensions of the base log point).\\
The only result we will need later on is the following: 
\begin{prop}[({cor. \ref{logsp2}})]\label{logsp}
Let $X\to S$ be a proper and saturated morphism of log schemes,
  and let $Y\to X$ be a két covering. Let $s$ and $s'$ be two log points of $S$
  and assume that one has a specialization $\bar
    s'\to\bar{s}$ (where $\bar s$ and
  $\bar s'$ are some log geometric points over $s$ and
  $s'$). Let $p$ be the characteristic of $s$. One has a specialization morphism
\[\ggeom(Y_{s'}/s')^{(p')}\to\ggeom(Y_{s}/s)^{(p')}.\]

Moreover this morphism factorizes through
$\varprojlim_U\glog(Y_U)^{(p')}$, where $U$ runs through the két
neighborhoods of $\bar s$.\end{prop}

To prove this, our main result will
be the invariance of the log geometric fundamental group of an fs
log scheme $X$ saturated and of finite type over an fs log point $S$ with separably closed field by fs
base change that is an isomorphism on the underlying scheme. The
assumptions implies that our base change induces an isomorphism of
underlying schemes. Working étale locally on this scheme, we are reduced to
the case where this scheme is strictly local, where the log
geometric fundamental group can be explicitly described in terms of the
morphism of monoids $\overline M_{X}\to\overline M_S$.\\
Combining this base change invariance result with strict base change
invariance of the $(p')$-log geometric fundamental group and strict
specialization of the $(p')$-log geometric fundamental group (\cite{org2}),
we will get that if $X\to S$ is proper log smooth saturated morphism, and
$s_2,s_1$ are fs points of $S$ and $\bar s_2\to\bar s_1$ is a
specialization of log geometric points of $S$ over $s_2$ and $s_1$, then
there is a specialization morphism
$\ggeom(X_{s_2})\to\ggeom(X_{s_1})$ (this is the only result we will really
need in the following).\\
 
\begin{lem}\label{lemchgmtbase} Let $s'\to s$ be a strict morphism of log points such that
  $\mring s'$ and $\mring s$ are geometric points of characteristic $p$. Let $X\to s$ be a
  morphism of fs log schemes such that $\mring X\to\mring s$ is of finite type.\\
Then $F:\KCov(X)^{(p')}\to \KCov(X_{s'})^{(p')}$ is an equivalence of categories.\end{lem}
\dem
If $T$ is a connected két covering of $X$, $\mring{T\times_ss'}\to\mring
T\times_{\mring s}\mring s'$ is an isomorphism since $s'\to s$ is
strict. $\mring T\times_{\mring s}\mring s'$ is connected too, so we get
that the functor $F$ is fully faithful.\\
As one already knows that $F$ is fully faithful for any $X$, and as strict
étale surjective morphisms satisfy effective descent for két coverings, one may prove
the result étale locally, and thus assume that $X$ has a global chart
$X\to\Spec \mbf Z[P]$.\\
Let $S'$ be a két covering of $X_{s'}$. Then there exists a $(p')$-Kummer
morphism of monoids $P\to Q$ such that \[S'_Q:=S'\times_{\Spec \mbf
  Z[P]}\Spec\mbf Z[Q]\to X_{s',Q}:=X_{s'}\times_{\Spec \mbf
  Z[P]}\Spec\mbf Z[Q]\] is strict étale (and surjective).\\
But, since $\mring X_{s',Q}\to\mring X_Q\times_{\mring s}\mring s'$ is an
isomorphism of schemes, $\Covalg(\mring X_{s',Q})\to\Covalg(\mring X_Q)$ is an equivalence of
categories~(\cite[cor 4.5]{org}). Thus, there is a strict étale covering
$S_Q$ of $X_Q$ (and thus $S_Q\to X$ is a két covering) such that $S'_Q$ is
$X_{s',Q}$-isomorphic to $S_Q\times_ss'$.\\
Thus $F$ is an equivalence of categories.
\findem

Let now $s'\to s$ be a morphism of fs log points, such that the underlying
morphism of schemes $\mathring s'\to
\mathring s$ is an isomorphism of geometric points, and let $X\to s$ be a
saturated morphism of fs log schemes with $\mathring X$
nœtherian and $\mathring{X}\to\mring{s}$ geometrically connected. Since
$X\to s$ is saturated, it is log geometrically connected.\\
Let $\bar x'$ be a log geometric point of $X'=X\times_ss'$ and let $\bar
x$, $\bar s'$ and $\bar s$ be its image in $X$, $s'$ and $s$ respectively.\\
We have a commutative diagram
\[\xymatrix{\glog(X',\bar x') \ar[r] \ar[d] & \glog(X,\bar x) \ar[d]\\
\glog(s',\bar s)\ar[r] & \glog(s,\bar s)}\]

\begin{thm}\label{chgmtbase1} The map $\ggeom(X'/s',\bar x')\to\ggeom(X/s,\bar x)$ is an isomorphism.\end{thm}
\dem
Let $(s_i,\bar s_i)_{i\in I}$ be a cofinal system of pointed Galois
connected két coverings of $(s,\bar s)$. Let $\tilde s_i$ be the
reduced subscheme of $s_i$ endowed with the inverse image log structure. Let us write $(X_i,\bar x_i)=(X\times_s\tilde s_i,\bar x\times_{\bar
  s}\bar s_i)$.\\
Let $(s'_j,\bar s'_j)_{j\in J}$ be a cofinal system of pointed Galois
connected két coverings of $(s',\bar s')$. Let $\tilde s'_j$ be the
reduced subscheme of $s'_j$ endowed with the inverse image log
structure. Let us write $(X'_j,\bar x'_j)=(X\times_{s'}s'_j,\bar
x'\times_{\bar s'}\bar s'_j)$.\\
One has to prove that
 \[\varprojlim_j\glog(X'_j,\bar
 x'_j)\to\varprojlim_i\glog(X_i,\bar
x_i)\]
is an isomorphism.\\

If $Y$ is an fs log scheme, $\epsilon_Y:Y_{\ket}\to \mathring
Y_{\etale}$ denotes the usual morphism of topoi from the két topos of $Y$
to the étale topos
of the underlying scheme of $Y$.\\
The log scheme $\tilde s_i$ has the same underlying scheme as $s$, so $\mring X_i\to\mring
X$ is an isomorphism of schemes since $X\to s$ is assumed to be saturated.
For the same reason, $\mring X'_j\to \mring X'$ and $\mring X'\to\mring X$ are also isomorphisms of schemes.\\
More precisely, for any $i$ there is $j_0$ such that for $i\geq i'$ and $j\geq j'\geq j_0$, one
has a 2-commutative diagram:
\[\xymatrix {
    X'_{j,\ket} \ar[rr] \ar[dd] \ar[dr] && X'_{j',\ket} \ar[dr] \ar[dd] |\hole \\
    & X_{i,\ket} \ar[rr] \ar[dd] && X_{i',\ket} \ar[dd] \\
    \mathring X'_{j,\etale} \ar[rr] |\hole \ar[dr] && \mathring X'_{j',\etale} \ar[rd] \\
    & \mathring X_{i,\etale} \ar[rr] && \mathring X_{i',\etale} \\
  }\]
where all the morphisms of schemes of the bottom square are isomorphisms.\\
If $G$ is a finite group, one thus has 2-commutative diagrams of stacks over $\mathring
X_{\etale}$:
\[\xymatrix{\epsilon_{X'_{j'}*}\Tors_{X'_{j',\ket}}(G) \ar[r]\ar[d] & \epsilon_{
    X'_{j}*}\Tors_{X'_{j,\ket}}(G) \ar[d]\\
  \epsilon_{X_{i'}*}\Tors_{X_{i',\ket}}(G) \ar[r] &
  \epsilon_{X_{i}*}\Tors_{X_{i,\ket}}(G)} \]
More precisely, let us write $IJ=I\coprod J$ endowed with the partial order such that:
\begin{itemize}
\item the restriction of the partial order to $I$ (resp. $J$) is the usual
  one,
\item if $i\in I$ and $j\in J$, then $i\leq j$ if and only if there exists a (necessarily unique)
morphism of pointed fs log schemes $(s'_j,\bar s'_j)\to (s_i,\bar s_i)$
that makes the square \[\begin{array}{ccc}(s'_j,\bar s'_j) & \to &
  (s_i,\bar s_i)\\ \dar & & \dar \\ (s',\bar s') & \to & (s,\bar
  s)\end{array}\] commutative,
\item if $j\in J$ and $i\in I$, then $j\nleq i$.\end{itemize}

Let us also denote by $IJ$ the corresponding category.
Then one has a fibered category over $IJ^{\op}\times \mring X_{\etale}$, whose
fiber in $(i,U)$ is $\epsilon_{X_{i}*}\Tors_{X_{i,\ket}}(G)(U)$ and the
fiber in $(j,U)$ is $\epsilon_{X'_{j}*}\Tors_{X'_{j,\ket}}(G)(U)$.\\

By taking the inductive limit over $i\in I$, which is directed, one
gets, according to~\cite[I.1.10]{giraud}, a fibered category over
$\mathring X_{\etale}$ whose fiber in $U$ is $\injLim_i
\epsilon_{X_{i}*}\Tors_{X_{i,\ket}}(G)(U)$; let us
denote by $\injLim_i \epsilon_{X_{i}*}\Tors_{
  X_{i,\ket}}(G)$ the stack associated to this fibered category (and we do
the same for $X'$ and $J$).\\
Since $\mring X$ is assumed to be noetherian (and thus $\mring X_{\et}$ is a
coherent topos), the functor from  $\injLim_j \Tors(G,
X'_{j,\ket})$ to the category of global sections of $\injLim_j\epsilon_{X'_j*}\Tors_{
  X'_{j,\ket}}(G)$ is an equivalence of categories. Moreover \[\injLim_j \Tors(G,
X'_{j,\ket})\simeq\injLim_j \Tors(G,\glog(X'_j,\bar x'_j)-\Ens)\simeq\Tors(G,\varprojlim \glog(X'_j,\bar x'_j)-\Ens).\]
The 2-commutative diagrams induce a morphism of stacks:
\begin{equation}\label{morphisme_champ}\injLim_j\epsilon_{X'_j*}\Tors_{X'_{j,\ket}}(G)\to\injLim_i\epsilon_{X_i*}\Tors_{X_{i,\ket}}(G)\end{equation}

We thus have to prove that~(\ref{morphisme_champ}) is an equivalence of
stacks, which can be proved on the fibers (since $\mring X_{\et}$ has
enough points;~\cite[cor. III.2.1.5.8]{giraud}). This does not depend on the base points anymore (we thus
may forget those, so that we may use the notations again to denote other points).\\
Let $x$ be a point of $\mring X_{\et}$, $\bar x'$ a point of $X'_{\ket}$ above $x$ and $\bar s'$
with image $\bar x$ in $X_{\ket}$. Let $\bar x_i=(\bar x,\bar s)$ and let $\bar x'_j=(\bar x',\bar s'_j)$. let $V_x$ be the category of
étale neighborhoods of $x$ in $\mring X$. Then one has (by coherence of the
morphism of topoi $\epsilon_{X_i}$, as in~\cite[proof of 2.4]{org2}):
\[\begin{array}{rl}\injLim_{U\in
    V_x}\injLim_i\epsilon_{X_i*}\Tors_{X_{i,\ket}}(G)(U) &
  =\injLim_i\injLim_{U\in V_x}\epsilon_{X_i*}\Tors_{X_{i,\ket}}(G)(U)\\
 & =\injLim_i\Tors(G,X(x)_{i,\ket})\\
 & =\Tors(G,\varprojlim_i\glog(X(x)_{i,\ket},\bar x_i)-\Ens),\end{array}\]
and one has a similar result for $X'$.\\
One thus only have to prove that \[\varprojlim_j\glog(X'(x)_{j,\ket},\bar
x'_j)\to\varprojlim_i\glog(X(x)_{i,\ket},\bar
x_i)\] is an isomorphism.\\
We are thus reduced to the case where $\mathring X$ is a strictly local and
noetherian scheme.
But then (\cite[prop. 3.1.11]{stix}), for $\mathring X$ a strictly local and
noetherian scheme, 
\[\begin{array}{rcl}\varprojlim_i\glog(X_{i,\ket},\bar x_i) & = & \varprojlim_i \Hom(\overline M_{X_i,x_i}^{\gp}, \widehat{\mbf
  Z}^{(p')}) \\ & = & \Hom(\varinjlim_i \overline M_{X_i,x_i}^{\gp},\widehat{\mbf Z}^{(p')})
\\ & = & \Hom(\Coker(\overline M_s^{\gp}\to \overline M_{X,x}^{\gp}),\widehat{\mbf Z}^{(p')}),\end{array}\]
and one has a similar result for $X'$.\\
Since $\overline M_{X',x'}=\overline M_{X,x}\oplus_{\overline M_s}\overline M_{s'}$, one has $\overline M_{X',x'}^{\gp}=\overline M_{X,x}^{\gp}\oplus_{\overline M_s^{\gp}}\overline M_{s'}^{\gp}$. Thus, $\Coker(\overline M_s^{\gp}\to \overline M_{X,x}^{\gp})\to\Coker(\overline
M_{s'}^{\gp}\to \overline M_{X',x'}^{\gp})$ is an isomorphism.\\
One thus gets the wanted result.
\findem

Let us assume now that $(s',\bar s')\to (s,\bar s)$ is simply assumed to be a morphism of
log points, that $Y\to s$ is a saturated morphism and that $X\to Y$ is a két
morphism with $\mring X$ of finite type over $s$.
\begin{cor}\label{loginvar}The map of profinite
  groups \[\ggeom(X/s,\bar x)^{(p')}\to\ggeom(X'/s',\bar x')^{(p')}\] is an
  isomorphism.\end{cor}
\dem
By replacing $s$ (resp. $s'$) by the closed reduced subscheme of a connected
két covering of $s$ (resp. $s'$), one can assume that $X\to s$ is saturated
($\mring X\to\mring s$ will still be of finite type).\\
If $(t,\bar t)\to (s,\bar s)$ is a strict étale covering, then $\ggeom(X_t/t,\bar
x_t)\to\ggeom(X/s,\bar x)$ is an isomorphism. Thus, by writing $s_0$ for the
separable closure of $s$ and by taking the
projective limit over pointed strict étale coverings (since
$\glog(X_{s_0})=\varprojlim\glog(X_t)$, where $t$ runs through pointed
strict étale coverings of $s$), one gets that $\ggeom(X_{s_0}/s_0,\bar
x_0)\to\ggeom(X/s,\bar x)$ is an isomorphism.\\
One thus may assume that $\mring s$ and $\mring s'$ are geometric points.\\
Let us consider the fs log scheme $s''$ whose underlying scheme is $\mring
s'$ and whose log structure is the inverse image of the log structure of
$s$.\\
Thus, one has morphisms $s'\to s''\to s$, where $s'\to s''$ is an
isomorphism on the underlying schemes and $s''\to s$ is strict.\\
But according to lemma~\ref{lemchgmtbase}, $\glog(X_{s''})^{(p')}\to
\glog(X)^{(p')}$ and $\glog(s'')^{(p')}\to\glog(s)^{(p')}$ are isomorphisms. Thus,
\[\ggeom(X_{s''}/s'')^{(p')}\to\ggeom(X/s)^{(p')}\] is an isomorphism.\\
By~\ref{chgmtbase1}, $\ggeom(X_{s'}/s')^{(p')}\to\ggeom(X_{s''}/s'')^{(p')}$ is also an
isomorphism.
\findem

\begin{dfn}Let $S$ be a fs log scheme such that $\mring S$ is strictly local. Let $Y\to S$ be a morphism of fs log schemes whose special fiber is log geometrically connected.
The log geometric fundamental group of $Y$ over $S$ is \[\ggeom(Y_S/S):=\varprojlim_{T} \glog(Y_T),\] where $T$ runs
through pointed két coverings of $(S,\bar s)$.\end{dfn} 
Recall that if $S$ is a strictly local scheme
  with closed point $s$ and $X$ is a connected fs log scheme such that
  $\mring X$ is proper over $S$, then
\[\KCov(X)\to\KCov(X_s)\]is an equivalence of categories (proposition~\ref{orgsp}).\\
Let $X\to S$ be a proper and saturated morphism of log schemes,
  and let $Y\to X$ be a két covering. Let $s$ and $s'$ be two log points of $S$
  and assume that one has a specialization $\bar
    s'\to\bar{s}$ (where $\bar s$ and
  $\bar s'$ are some log geometric points over $s$ and
  $s'$).
Let $Z$ be the strictly local scheme of $S$ at $s$ endowed with the inverse
image log structure, and let $z$ be its closed point, endowed with the
inverse image log structure.
One also has an isomorphism $\ggeom(Y_Z/Z):=\varprojlim_{U}\glog(Y_U)$, where
$U$ runs through két neighborhoods of $\bar s$ in $S$.\\
One has the following arrows (defined up to inner homomorphisms):
\[\ggeom(Y_{s}/s)^{(p')}\stackrel{\simeq}{\to}\ggeom(Y_z/z)^{(p')}\stackrel{\simeq}{\to}\ggeom(Y_Z/Z)^{(p')}\leftarrow\ggeom(Y_{s'}/s')^{(p')}\]
where the first two homomorphism are isomorphisms according to
corollary~\ref{loginvar} and theorem~\ref{orgsp}.\\
\begin{cor}\label{logsp2}One has a specialization morphism
\[\ggeom(Y_{s'}/s')^{(p')}\to\ggeom(Y_{s}/s)^{(p')}\] that factors through $\ggeom(Y_Z/Z)^{(p')}$.\end{cor}


\section{Cospecialization of tempered fundamental groups}

\subsection{Topological cospecialization of semistable curves}

Let $f:X\to Y$ be a semistable curve, and let $\bar y_2\to \bar y_1$ be a specialization of
geometric points of $Y$. In this section we will define a cospecialization
map of graphs $\mbb G(X_{\bar y_1})\to \mbb G(X_{\bar y_2})$.\\

\begin{lem}Assume $Y$ is strictly local of special point $y_1$, and $X\to
  Y$ strictly semistable. Let $x$
  be a node or a generic point of $X_{y_1}$. Then $X(x)_{y_2}$ is either
  contained in the smooth locus of a geometrically irreducible component $F(x)$ of
  $X(x)_{y_2}$ or contains a single node $F(x)$ of $X(x)_{y_2}$, which is rational.\\
\end{lem}
\dem
By replacing $Y$ by a closed subscheme, one can assume that $y_2$ is the
generic point of $Y$ and $Y$ is integral.\\
\begin{enumerate}[(i)]
\item
If $x$ is in the smooth locus of $X_{y_1}$, $X\to Y$ is smooth at $x$, and
$X(x)_{y_2}$ is geometrically connected by local 0-acyclicity of smooth
morphisms.
\item
If $x$ is a node, one can assume that $Y=\Spec A$ and $f$ factorizes
through an étale morphism $X\to\Spec B$ with $B=A[u,v]/uv-a$ and
$a(y_1)=0$.\\
If $a=0$ let $Z=X\times_{\Spec B}\Spec A$ where $g:B\to A$ is defined by
$g(u)=g(v)=0$ (this is the closed subscheme of $X$ defined by the
node; in particular $Z_{y_2}$ is the union of all the nodes of $X_{y_2}$). $Z\to Y$ is étale and thus $Z(x)\to Y$ is an isomorphism. Thus
$Z(x)_{y_2}$ is just a rational point $F(x)$.\\
If $a\neq 0$, $a(y_2)\neq 0$ and thus $X_{y_2}$  is smooth. Since $X\to Y$
is a semistable curve, it is separable (\ie flat with separable geometric
fibers). By applying \cite[cor. 18.9.8]{ega4} to $X(x)\to Y$, one gets that $X(x)_{y_2}$ is
geometrically connected.
\end{enumerate}
\findem
Let $x$ be a point of $X_{\bar y_1}$. If $F(x)$ is a rational node of $X_{y_2}$, then it defines an edge $F_0(x)$ of $\mbb G_{\bar y_2}$. If $F(x)$ is a geometrically irreducible component of $X_{y_2}$, then it defines a vertex $F_0(x)$ of $\mbb G_{\bar y_2}$.
\begin{lem}
If $\phi:X'\to X$ is a finite open morphism of strictly semistable curves over
$Y$ which maps nodes to nodes on every fiber, then $\phi F'=F\phi$.\end{lem}
\dem Indeed
$\phi(x)$ is in the closure of $\phi F'(x)$, thus $F\phi(x)$ is in the
closure of $\phi F'(x)$. One only has to prove that if $F\phi(x)$ is a
node, then $\phi F'(x)$ is also a node. Let us assume that  $F \phi(x)$
is a node of $X_{y_2}$. Let $z_1$ and $z_2$ be the two generic points of
the irreducible components of $X_{y_2}$ whose closures contain $F \phi(x)$
(and thus also $\phi(x)$). Since $\phi$ is open, there exists $z'_1$ and
$z'_2$ in $X(x)_{y_2}$ such that $\phi(z'_1)=z_1$ and
$\phi(z'_2)=z_2$. Thus $X(x)_{y_2}$ cannot be in a single irreducible
component of $X_{y_2}$, and thus $F'(x)$ is a node of $X'_{y_2}$. By
assumption, $\phi F'(x)$ is a node of $X_{y_2}$.\findem


\begin{cor}\label{cosptop}
There is a unique generalized morphism of graphs \[\psi:\mbb G(X_{\bar
  y_1})\to\mbb G(X_{\bar y_2})\] which is
\begin{itemize}
\item functorial for étale morphisms $X'\to X$,
\item compatible with base change $Y'\to Y$,
\item such that if $f:X\to Y$ is strictly semistable and $Y$ is strictly
  local with special point $\bar y_1$, $\psi(x)=F_0(x)$ for any
node or generic point $x$.
\end{itemize}
\end{cor}
\dem
Let $f:X\to Y$ be a strictly semistable curve, and let $\bar y_2\to \bar y_1$ be a specialization of
geometric points of $Y$. After replacing $Y$ by its strict localization at $\bar y_1$, onne can assume that $Y$ is strictly local and $\bar y_1$ is the closed point.\\
If $e$ is a vertex of $\mbb G(X_{\bar y_1})$, then $\psi(e):=F_0(x)$ where $x$ is the node of $X_{\bar y_1}$ corresponding to $e$.
If $v$ is a vertex of $\mbb G(X_{\bar y_1})$, then $\psi(v):=F_0(x)$ where $x$ is the generic point of the irreducible component of $X_{\bar y_1}$ corresponding to $e$. Let $b$ be a branch
  of an edge $e$ in $\mbb G(X_{y_1})$ that abuts to a vertex $v$. Then $F(x)\subset F(y)$, where $x$ is the node corresponding to $e$ and $y$ is the generic point of the irreducible component corresponding to $v$. If $F(x)=F(y)$, then $\psi(b):=F_0(x)=F_0(y)$. Otherwise, $\psi(e)$ is an edge and $\psi(v)$ is a vertex, and there is a branch $b'$ of $\psi(e)$ abutting to $\psi(v)$. Since $X_{\bar y_2}$ is strictly
  semistable), this branch is unique. Let $\psi(b)=b'$.\\
This morphism is clearly compatible with étale morphisms of strictly
semistable curve.\\

If $X\to Y$ is now a general semistable curve, one choses an étale
covering family $X'\to X$, and let $X''=X'\times_XX'$.\\
One has a commutative diagram:
\[\begin{array}{ccccc} \mbb G(X''_{\bar y_1}) & \rightrightarrows & \mbb
  G(X'_{\bar y_1}) & \to & \mbb G(X_{\bar y_1})\\ \dar & & \dar & & \\
\mbb G(X''_{\bar y_2}) & \rightrightarrows & \mbb
  G(X'_{\bar y_2}) & \to & \mbb G(X_{\bar y_2})\end{array}\]
There is a unique generalized morphism of graphs $\psi:\mbb G(X_{\bar
  y_1})\to\mbb G(X_{\bar y_2})$ making the diagram commutative.
\findem

The cospecialization generalized morphisms of graphs are also compatible
with finite open morphisms of strictly semistable curves that maps nodes to
nodes fiberwise.\\

We want to know when this generalized morphism of graphs is an isomorphism.


\begin{prop}\label{isomproper}
If $\psi:\mbb G(X_{\bar y_1})\to\mbb G(X_{\bar y_2})$ is a genuine morphism
of graphs and $f$ is proper, then $\psi$ is an isomorphism.
\end{prop}
\dem
One may assume $Y=\Spec A$ to be strictly local and integral with special
point $y_1$ and generic point $y_2$.\\
The assumption means that étale locally on the special fiber (and thus on
$X$ by properness), $X$ is isomorphic to $\Spec A[u,v]/uv$ or is smooth.\\
 
Let $Z\subset X$ be the non smooth locus
of $X\to Y$, endowed with the reduced subscheme structure. $Z\to Y$ is
étale (as can be seen étale locally over $X$), and proper. One thus gets
that $F$ induces a bijection between nodes of $X_{\bar y_1}$ and $X_{\bar
  y_2}$.\\
Let $\widetilde X$ be the blowup of $X$ along $Z$. 
When $X=\Spec A[u,v]/(uv)$, $Z$ is defined by the ideal generated by $u$
and $v$, and $\widetilde X=\Spec A[u]\coprod \Spec A[v]$.\\
Thus by looking étale locally over $X$, one sees
that $\widetilde X$ is smooth over $Y$, and that $\widetilde X_y$ is simply the
normalization of $X_y$.\\
Since we assumed $X\to Y$ to be proper, $\widetilde X\to Y$ is smooth and
proper, thus its Stein factorization induces a bijection between the
connected components of $\widetilde X_{\bar y_1}$ and $\widetilde X_{\bar
  y_2}$, and thus the map between the irreducible components of $
X_{\bar y_1}$ and $X_{\bar y_2}$ is a bijection too.
\findem



\begin{prop}\label{invstrates}
Let $f:X\to Y$ be a log semistable curve and let $\bar y_2\to
\bar y_1$ be a specialization of log geometric point.\\
Assume $\overline M_{\bar y_1}\to\overline M_{\bar y_2}$ is an isomorphism.
Then $\psi:\mbb G(X_{y_1})\to\mbb G(X_{y_2})$ is a genuine morphism of
graphs. \end{prop}
\dem
One can assume $Y$ to be strictly local, integral with generic point $y_2$:
$Y=\Spec A$, with a chart $P\to A$.\\

To show that it is a genuine morphism, one only has to prove that $\psi(e)$
is an edge if $e$ is an edge of $\mbb G(X_{y_1})$. This is not changed by
an étale morphism, so that one can simply assume $X=\Spec A\otimes_{\mbf
  Z[P]}\mbf Z[Q]$ with $Q=(P\oplus<u,v>)/(u+v=p)$ and $p\in P$, such that
the image of $p$ in $M_{\bar y_1}$ is not invertible. Thus the image of $p$
in $M_{\bar y_2}$ is not invertible and thus $X=\Spec A[u,v]/(uv=0)$, which
gives the wanted result.\\
\findem


\subsection{Topological cospecialization and két morphisms}
\begin{prop}Let $f:X\to Y$ be a proper log semistable curve. Let $\bar y_2\to\bar y_1$
be a specialization of log geometric points. Let $S_{\bar y_1}$ be a log geometric k\'et covering of $X_{y_1}/\bar y_1$. Let $S_{\bar y_2}$ be the corresponding log geometric covering of $X_{y_2}/\bar y_2$ defined by proposition \ref{logsp}. Then there is a canonical generalized morphism of graphs
\[\mbb G(S_{\bar y_1}) \to \mbb G(S_{\bar
  y_2}).\]
\end{prop}
\dem
According to proposition~\ref{logsp}, $S_{\bar y_1}$ extends to a két neighborhood $U$ of $\bar y_1$ in a két covering
$S\to X_U$. After further két localization, one may even assume $S\to X_U$
to be a log semistable curve.\\
Thus one can use corollary~\ref{cosptop} and get a generalized morphism of
graphs $\mbb G(S_{\bar y_1})\to\mbb G(S_{\bar y_2})$. It does not depend on
the choice of $U$ or $S$ since any two extensions of $S_0$ to a két
neighborhood of $\bar y_1$ are isomorphic over a smaller két neighborhood
and this isomorphism is unique up to further két localization.\findem
If $S'_{\bar y_1}\to S_{\bar y_2}$ is a morphism of log geometric két coverings, then the following diagram is commutative:
\[\begin{array}{ccc} \mbb G(S'_{\bar y_1}) & \to  & \mbb G(S'_{\bar y_2})\\
\dar & & \dar \\ \mbb G(S_{\bar y_1}) & \to & \mbb G(S_{\bar
  y_2})\end{array}\]
If $\overline M_{Y,y_1}\to\overline M_{Y,y_2}$ is an isomorphism,
$\overline M_{U,y_1}\to\overline M_{U,y_2}$ is still an isomorphism, so
that one can still apply proposition~\ref{invstrates} to $S$: the morphism $\mbb G(S_{\bar y_1}) \to \mbb G(S_{\bar
  y_2})$ is a genuine morphism of graphs.\\
 
\subsection{Cospecialization of tempered fundamental groups}
Let $Y\to O_K$ be a morphism of fs log schemes.\\
Let $Y_{\tr}$ be the open locus of $Y$  where the log structure is trivial
($Y_{\tr}\subset Y_\eta$). Let $\fk Y$ be the completion of $Y$ along its
closed fiber. Then $\fk Y_\eta$ is an analytic domain of $Y^{\an}$.
Let $Y_0=\fk Y_\eta\cap Y_{\tr}^{\an}\subset Y^{\an}$.\\
Let $\tilde y$  be a $K'$-point of $Y_0$ where $K'$ is a complete extension of $K$. One has a canonical morphism of log schemes $\Spec O_{K'}\to Y$ where $\Spec O_{K'}$ is endowed with the log structure given by $O_{K'}\backslash\{0\}\to O_{K'}$. The \emph{log reduction} $\tilde s$ of $\tilde y$ is the log point of $Y$
corresponding to the special point of $\Spec O_{K'}$ with the inverse image of the log structure of $\Spec O_{K'}$. If $K'$ has discrete valuation, then $\tilde s$ is a fs log point. If $K'$ is algebraically closed, $\tilde s$ is a geometric log point.\\

\begin{dfn} The category $\widetilde\Pt^{\an}(Y)$ is the category whose objects are the geometric
  points $\bar y$ of $Y_0$ such that $\mcal H(y)$ is discretely valued (where $y$
  is the underlying point of $\bar y$) and $\Hom_{\Pt^{\an}(Y)}(\bar y,\bar
  y')$ is the set of két specializations in $Y_k$ from the log reduction
  $\bar s$ of $\bar y$ to the log reduction $\bar s'$ of $\bar y'$ such that there  exists some specialization $\bar y\to \bar y'$ of geometric points in the sense of algebraic étale topology  for which the following diagram commutes:
\[\xymatrix{ \bar y\ar[r]\ar[d] & \bar s\ar[d]\\ 
\bar y'\ar[r] & \bar s'
}\]
The category $\Pt^{\an}_0(Y)$ is the category obtained from $\widetilde\Pt^{\an}(Y)$ by inverting the class of morphisms $\bar y\to \bar y'$ such that $\overline M_{Y,\bar s'}\to\overline M_{Y,\bar s}$ is an isomorphism.
\end{dfn}
Let $\OutGptop $ be the category of topological groups with outer morphisms.

\begin{thm}\label{cospcourbes}Let $O_K$ be a complete discretely valued field of residue characteristic $p\geq 0$, let $\mbb L$ be a set of integers such that $p\notin L$. Let $Y\to O_K$ be a morphism of fs log schemes and $X\to Y$ be a proper log
semistable curve. Let $U$ be the open locus of $X$ where the log structure is trivial. Then
there is a functor $\gtemp(U_{(\cdot)}):\Pt^{\an}_0(Y)^{\op}\to\OutGptop $
sending $\bar y$ to $\gtemp(U_{\bar y})$.
\end{thm}
\dem
Let $\bar y_2\to\bar y_1$ be a morphism $\widetilde\Pt^{\an}(Y)$. One has to construct a cospecialization morphism $\gtemp(X_{\bar y_1})\to\gtemp(X_{\bar y_2})$.\\
One has a cospecialization functor
\[F:\KCovgeom(X_{s_1}/s_1)^{\mbb L}\to\KCovgeom(X_{s_2}/s_2)^{\mbb L}\] which factors through
$\KCovgeom(X_{T}/T)^{\mbb L}$ where $T$ is the strict localization at $s_1$.\\
The cospecialization functor
$\KCovgeom(X_{s_i}/s_i)^{\mbb L}\to\Covalg(U_{\bar y_i})$ is an equivalence since $y_i\in Y_{\tr}$ (\ref{speclogdvrgeom}). If one choses a specialization $\bar y_2\to\bar y_1$ above $\bar s_2\to\bar s_1$, then one can apply \cite[cor. XIII.2.9]{sga} to $U_K\subset X_K\to Y_K$: one gets that the functor
$\Covalg(U_{\bar y_i})^{\mbb L}\to\Covalg(U_{\bar y_2})^{\mbb L}$ is
also an equivalence. Thus $F$ is an
equivalence.\\
Let $S_{\bar s_1}$ be a log geometric
két covering of $X_{\bar s_{1}}$ and let $S_{\bar s_2}$ (resp. $S_1$, $S_2$)
be the corresponding covering of $X_{\bar s_{2}}$ (resp. $U_{\bar
  y_1}$, $U_{\bar y_2}$).\\


There are maps (functorially in $S$):
\[|S^{\an}_{\bar s_1}|\leftarrow |\mbb G(S_{\bar s_1})|\to |\mbb G(S_{\bar s_2})|\to
|S^{\an}_{\bar s_2}|\]
where the first and third map are  the embedding of the skeleton of an
anlytic curve. They are homotopy equivalences.\\
One thus get a morphism of homotopy types
$|S^{\an}_{\bar s_1}|\to|S^{\an}_{\bar s_2}|$ functorially in $S$. According to propositions \ref{invstrates} and \ref{isomproper}, if $\overline M_{Y,\bar s_1}\to\overline M_{Y,\bar s_2}$ is an isomorphism, $|S^{\an}_{\bar s_1}|\to|S^{\an}_{\bar s_2}|$ is an isomorphism of homotopy types.\\
With the notations of proposition~\ref{stackydfn}, one thus gets a functor of fibered categories:
\[\begin{array}{ccc} \Dtop(U_{\bar y_2}) & \to & \Dtop(U_{\bar y_1})\\
\dar & & \dar\\
 \Covalg(U_{\bar y_{2}})^{\mbb L} & \simeq & \Covalg(U_{\bar
    y_{1}})^{\mbb L}\end{array}\]
Using proposition~\ref{stackydfn}, this induces a functor of associated stacks:
\[\begin{array}{ccc} \Dtemp(U_{\bar y_2})^{\mbb L} & \to & \Dtemp(U_{\bar
    y_1})^{\mbb L}\\
\dar & & \dar\\
 \Covalg(U_{\bar y_{2}})^{\mbb L} & \simeq & \Covalg(U_{\bar
    y_{1}})^{\mbb L}\end{array}\]
By taking the global sections one gets a functor:
\[\Covtemp(U_{\bar y_2})^{\mbb L}\to\Covtemp(U_{\bar y_1}))^{\mbb L},\]
which is an equivalence if $\overline M_{Y,\bar s_1}\to\overline M_{Y,\bar s_2}$ is an isomorphism.
It induces a cospecialization outer morphism of tempered fundamental groups
\[\gtemp(U_{\bar y_1})^{\mbb L}\to\gtemp(U_{\bar y_2})^{\mbb L},\]
which is an isomorphism if $\overline M_{Y,\bar s_1}\to\overline M_{Y,\bar s_2}$ is an isomorphism.
\findem

\begin{rem}
Such a functor cannot exist if $p\neq 0$ and $\mbb L$ is the set of all
primes. Indeed, if $X_1$ and $X_2$ are two Mumford curves with isomorphic
stable reduction but with different metrics on the graphs of their stable
models, then their tempered fundamental groups are not isomorphic~(\cite{metricmumford}). Let us
consider a moduli space of stable curves with level structure, endowed with its
canonical log structure, and a geometric point $\bar s$ in the special fiber of the moduli
space such that the corresponding stable curve has totally degenerate
reduction. In particular, it has at least two double points, and thus
the rank of $\overline M_{\bar s}^{\gp}$ is at least two. Let us take two
valuative fs log points $s_1$ and $s_2$ (\emph{i.e.} $\overline
M_{s_i}\simeq\mbf N$) such that the corresponding morphisms $\overline M^{\gp}_{\bar
  s}\to\mbf Z$ are not collinear. Let $\eta_1$ and $\eta_2$ be
discretely valued points of the
analytic geometric fiber whose log reductions are $s_1$ and $s_2$. Then the
two corresponding geometric Mumford curves have different metric on the
graph of their stable model, and thus have non isomorphic tempered
fundamental groups. But two geometric log points over $s_1$ and $s_2$ are
isomorphic with respect to specialization for két topology.
\end{rem}


\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\providecommand{\href}[2]{#2}
\begin{thebibliography}{And03b}

\bibitem[And03a]{andre2}
Yves André, \emph{On a geometric description of
  {$\mathrm{Gal}({\overline{\mathbf {Q}}}_p/{\mathbf {Q}}_p)$} and a $p$-adic
  avatar of {$\widehat{GT}$}}, Duke Mathematical Journal \textbf{119} (2003),
  1--39.

\bibitem[And03b]{andre1}
\bysame, \emph{Period mappings and differential equations : From {$\mathbf C$}
  to {$\mathbf C_p$}}, MSJ Memoirs, vol.~12, Mathematical Society of Japan,
  Tokyo, 2003.

\bibitem[Ber90]{berk}
Vladimir~G. Berkovich, \emph{Spectral theory and analytic geometry over
  non-{A}rchimedean fields}, Mathematical Surveys and Monographs, vol.~33,
  American Mathematical Society, Providence, 1990.

\bibitem[dJ95]{dJ1}
Aise~Johan de~Jong, \emph{{\'E}tale fundamental group of non-{A}rchimedean
  analytic spaces}, Compositio mathematica \textbf{97} (1995), 89--118.

\bibitem[GD67]{ega4}
Alexander Grothendieck and Jean Dieudonné, \emph{{Eléments de geométrie
  alg\'ebrique. IV: Etude locale des sch\'emas et des morphismes de sch\'emas
  (Quatrième partie)}}, Publications mathématiques de l'IHES \textbf{31}
  (1967), 5--361.

\bibitem[Gir71]{giraud}
Jean Giraud, \emph{cohomologie non abélienne}, Grundlehren der mathematischen
  Wissenschaften, vol. 179, Springer-Verlag, 1971.

\bibitem[Gro71]{sga}
Alexander Grothendieck (ed.), \emph{Revêtements étales et groupe fondamental
  ({SGA1})}, Lecture Notes in Mathematics, vol. 224, Berlin, Springer, 1971.

\bibitem[IKN05]{satmorph}
Luc Illusie, Kazuya Kato, and Chikara Nakayama, \emph{Quasi-unipotent
  logarithmic riemann-hilbert correspondences}, Journal of Mathematical
  Sciences of the University of Tokyo \textbf{12} (2005), 1--66.

\bibitem[Ill02]{ill}
Luc Illusie, \emph{An overview of the works of {K.} {Fujiwara}, {K.} {Kato},
  and {C.} {Nakayama} on logarithmic \'etale cohomology}, Cohomologies
  $p$-adiques et applications arithmétiques (II), Astérisque, vol. 279,
  Société mathématique de France, 2002, pp.~271--322.

\bibitem[Kat89]{kato}
Kazuya Kato, \emph{Logarithmic structures of {Fontaine}-{Illusie}}, Algebraic
  analysis,Geometry and Number theory, Proceedings of the JAMI inaugural
  conference (Baltimore and London), The John Hopkins University Press, 1989,
  pp.~191--224.

\bibitem[Kis00]{kisin}
Mark Kisin, \emph{Prime to $p$ fundamental groups and tame {Galois} action},
  Annales de l'Institut Fourier \textbf{50} (2000), no.~4, 1099--1126.

\bibitem[Lep]{cosp2}
Emmanuel Lepage, \emph{Coverings in $p$-adic analytic geometry and log
  coverings {II}: Cospecialization of the $(p')$-tempered fundamental group in
  higher dimensions}, arXiv:\ 0903:2349.

\bibitem[Lep10]{metricmumford}
\bysame, \emph{Tempered fundamental group and metric graph of a mumford curve},
  Publications of the Research Institute of Mathematical Sciences \textbf{46}
  (2010), 849--897.

\bibitem[Moc06]{mochi}
Shinichi Mochizuki, \emph{Semi-graphs of anabelioids}, Publications of the
  Research Institute of Mathematical Sciences \textbf{42} (2006), no.~1,
  221--322.

\bibitem[Ogu]{ogus}
Arthur Ogus, \emph{Lectures on logarithmic algebraic geometry}, notes
  pr\'eliminaires,
  \verb+http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf+.

\bibitem[Org]{org2}
Fabrice Orgogozo, \emph{Erratum et compl\'ements \`a l'article \guillemotleft\
  {A}lt\'erations et groupe fondamental premier \`a $p$ \guillemotright\ paru
  au bulletin de la s.m.f. (131), tome 1, 2003}, unpublished.

\bibitem[Org03]{org}
\bysame, \emph{Alt\'erations et groupe fondamental premier à $p$}, Bulletin de
  la société mathématique de France \textbf{131} (2003), no.~1, 123--147.

\bibitem[Sti02]{stix}
Jakob Stix, \emph{Projective anabelian curves in positive characteristic and
  descent theory for log-étale covers}, Ph.D. thesis, Universität Bonn
  Mathematisches Institut, Bonn, 2002.

\bibitem[Tsu97]{sattsuji}
Takeshi Tsuji, \emph{Saturated morphisms of logarithmic schemes}, 1997,
  unpublished.

\end{thebibliography}
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0909.2805.extracted_bib
\begin{thebibliography}{And03b}

\bibitem[And03a]{andre2}
Yves André, \emph{On a geometric description of
  {$\mathrm{Gal}({\overline{\mathbf {Q}}}_p/{\mathbf {Q}}_p)$} and a $p$-adic
  avatar of {$\widehat{GT}$}}, Duke Mathematical Journal \textbf{119} (2003),
  1--39.

\bibitem[And03b]{andre1}
\bysame, \emph{Period mappings and differential equations : From {$\mathbf C$}
  to {$\mathbf C_p$}}, MSJ Memoirs, vol.~12, Mathematical Society of Japan,
  Tokyo, 2003.

\bibitem[Ber90]{berk}
Vladimir~G. Berkovich, \emph{Spectral theory and analytic geometry over
  non-{A}rchimedean fields}, Mathematical Surveys and Monographs, vol.~33,
  American Mathematical Society, Providence, 1990.

\bibitem[dJ95]{dJ1}
Aise~Johan de~Jong, \emph{{\'E}tale fundamental group of non-{A}rchimedean
  analytic spaces}, Compositio mathematica \textbf{97} (1995), 89--118.

\bibitem[GD67]{ega4}
Alexander Grothendieck and Jean Dieudonné, \emph{{Eléments de geométrie
  alg\'ebrique. IV: Etude locale des sch\'emas et des morphismes de sch\'emas
  (Quatrième partie)}}, Publications mathématiques de l'IHES \textbf{31}
  (1967), 5--361.

\bibitem[Gir71]{giraud}
Jean Giraud, \emph{cohomologie non abélienne}, Grundlehren der mathematischen
  Wissenschaften, vol. 179, Springer-Verlag, 1971.

\bibitem[Gro71]{sga}
Alexander Grothendieck (ed.), \emph{Revêtements étales et groupe fondamental
  ({SGA1})}, Lecture Notes in Mathematics, vol. 224, Berlin, Springer, 1971.

\bibitem[IKN05]{satmorph}
Luc Illusie, Kazuya Kato, and Chikara Nakayama, \emph{Quasi-unipotent
  logarithmic riemann-hilbert correspondences}, Journal of Mathematical
  Sciences of the University of Tokyo \textbf{12} (2005), 1--66.

\bibitem[Ill02]{ill}
Luc Illusie, \emph{An overview of the works of {K.} {Fujiwara}, {K.} {Kato},
  and {C.} {Nakayama} on logarithmic \'etale cohomology}, Cohomologies
  $p$-adiques et applications arithmétiques (II), Astérisque, vol. 279,
  Société mathématique de France, 2002, pp.~271--322.

\bibitem[Kat89]{kato}
Kazuya Kato, \emph{Logarithmic structures of {Fontaine}-{Illusie}}, Algebraic
  analysis,Geometry and Number theory, Proceedings of the JAMI inaugural
  conference (Baltimore and London), The John Hopkins University Press, 1989,
  pp.~191--224.

\bibitem[Kis00]{kisin}
Mark Kisin, \emph{Prime to $p$ fundamental groups and tame {Galois} action},
  Annales de l'Institut Fourier \textbf{50} (2000), no.~4, 1099--1126.

\bibitem[Lep]{cosp2}
Emmanuel Lepage, \emph{Coverings in $p$-adic analytic geometry and log
  coverings {II}: Cospecialization of the $(p')$-tempered fundamental group in
  higher dimensions}, arXiv:\ 0903:2349.

\bibitem[Lep10]{metricmumford}
\bysame, \emph{Tempered fundamental group and metric graph of a mumford curve},
  Publications of the Research Institute of Mathematical Sciences \textbf{46}
  (2010), 849--897.

\bibitem[Moc06]{mochi}
Shinichi Mochizuki, \emph{Semi-graphs of anabelioids}, Publications of the
  Research Institute of Mathematical Sciences \textbf{42} (2006), no.~1,
  221--322.

\bibitem[Ogu]{ogus}
Arthur Ogus, \emph{Lectures on logarithmic algebraic geometry}, notes
  pr\'eliminaires,
  \verb+http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf+.

\bibitem[Org]{org2}
Fabrice Orgogozo, \emph{Erratum et compl\'ements \`a l'article \guillemotleft\
  {A}lt\'erations et groupe fondamental premier \`a $p$ \guillemotright\ paru
  au bulletin de la s.m.f. (131), tome 1, 2003}, unpublished.

\bibitem[Org03]{org}
\bysame, \emph{Alt\'erations et groupe fondamental premier à $p$}, Bulletin de
  la société mathématique de France \textbf{131} (2003), no.~1, 123--147.

\bibitem[Sti02]{stix}
Jakob Stix, \emph{Projective anabelian curves in positive characteristic and
  descent theory for log-étale covers}, Ph.D. thesis, Universität Bonn
  Mathematisches Institut, Bonn, 2002.

\bibitem[Tsu97]{sattsuji}
Takeshi Tsuji, \emph{Saturated morphisms of logarithmic schemes}, 1997,
  unpublished.

\end{thebibliography}