Finitely presentable, non-Hopfian groups with Kazhdan's Property~(T) and infinite outer automorphism group
Yves de Cornulier
We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of $p$-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.
math0502140.tex
%\title[Finitely presentable Kazhdan groups]
%{Finitely presentable, non-Hopfian groups with Kazhdan's
%Property~(T) and infinite outer automorphism group}
%\author[Cornulier]{Yves de Cornulier}



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\begin{document}
\title[Finitely presentable Kazhdan groups]
{Finitely presentable, non-Hopfian groups with Kazhdan's
Property~(T) and infinite outer automorphism group}

\author[Cornulier]{Yves de Cornulier}

\address{ \'Ecole Polytechnique F\'ed\'erale de Lausanne (EPFL)\\
Institut de G\'eom\'etrie, Alg\`ebre et Topologie (IGAT)\\
CH-1015 Lausanne, Switzerland}

\email{decornul@clipper.ens.fr}

\subjclass[2000]{Primary 20F28; Secondary 20G25, 17B56}
%20G25 Linear algebraic groups over local fields and their integers
%20F28 Automorphism groups of groups
%22E50 Representations of Lie and linear algebraic groups over local fields
%17B56 Cohomology of Lie (super)algebras

\tolerance=5000
% because of some overflows, it is need in order to make use of \linebreak[1]
% (otherwise the dangerous \linebreak is needed) - \allowbreak makes
% the previous line not aligned on the right so I do not use it.

\commby{Dan M. Barbasch}

\begin{abstract} We give simple examples of Kazhdan groups with
infinite outer automorphism groups. This answers a question of
Paulin, independently answered by Ollivier and Wise by completely
different methods. As arithmetic lattices in (non-semisimple) Lie
groups, our examples are in addition finitely presented.

We also use results of Abels about compact presentability of
$p$-adic groups to exhibit a finitely presented non-Hopfian
Kazhdan group. This answers a question of Ollivier and Wise.
\end{abstract}

\maketitle

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\section{Introduction}

Recall that a locally compact group is said to have Property~(T)
if every weakly continuous unitary representation with almost
invariant vectors\footnote{A representation $\pi:G\to
\mathcal{U}(\mathcal{H})$ almost has invariant vectors if for
every $\varepsilon>0$ and every finite subset $F\subseteq G$,
there exists a unit vector $\xi\in\mathcal{H}$ such that
$\|\pi(g)\xi-\xi\|<\varepsilon$ for every $g\in F$.} has nonzero
invariant vectors.

It was asked by Paulin in \cite[p.134]{HV} (1989) whether there
exists a group with Kazhdan's Property~(T) and with infinite outer
automorphism group. This question remained unanswered until 2004;
in particular, it is Question 18 in \cite{W}.

This question was motivated by the two following special cases.
The first is the case of lattices in {\it semisimple} groups over
local fields, which have long been considered as prototypical
examples of groups with Property~(T). If $\Gamma$ is such a
lattice, Mostow's rigidity Theorem and the fact that semisimple
groups have finite outer automorphism group imply that
$\textnormal{Out}(\Gamma)$ is finite. Secondly, a new source of
groups with Property~(T) appeared when Zuk \cite{Zuk} proved that
certain models of random groups have Property~(T). But they are
also hyperbolic, and Paulin proved \cite{Pau} that a hyperbolic
group with Property~(T) has finite outer automorphism group.

However, it turns out that various arithmetic lattices in
appropriate {\em non-semisimple} groups provide examples. For
instance, consider the additive group
$\textnormal{Mat}_{mn}(\mathbf{Z})$ of $m\times n$ matrices over
$\mathbf{Z}$, endowed with the action of
$\textnormal{GL}_n(\mathbf{Z})$ by left multiplication.

\begin{prop}
For every $n\ge 3$, $m\ge 1$,
$\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z})$ is a finitely presented linear
group, has Property~(T), is non-coHopfian\footnote{A group is
coHopfian (resp. Hopfian) if it is isomorphic to no proper
subgroup (resp. quotient) of itself.}, and its outer automorphism
group contains a copy of $\textnormal{PGL}_m(\mathbf{Z})$, hence
is infinite if $m\ge 2$.\label{p:Out_infini}
\end{prop}

We later learned that Ollivier and Wise \cite{OW} had
independently found examples of a very different nature. They
embed any countable group $G$ in $\text{Out}(\Gamma)$, where
$\Gamma$ has Property~(T), is a subgroup of a torsion-free
hyperbolic group, satisfying a certain ``graphical" small
cancellation condition (see also \cite{BS}). In contrast to our
examples, theirs are not, a priori, finitely presented; on the
other hand, our examples are certainly not subgroups of hyperbolic
groups since they all contain a copy of~$\mathbf{Z}^2$.

They also construct in \cite{OW} a non-coHopfian group with
Property~(T) that embeds in a hyperbolic group. Proposition
\ref{p:Out_infini} actually answers two questions in their paper:
namely, whether there exists a finitely presented group with
Property~(T) and without the coHopfian Property (resp. with
infinite outer automorphism group).

\begin{rem}Another example of non-coHopfian group with Property~(T) is
\linebreak[1]$\textnormal{PGL}_n(\mathbf{F}_p[X])$ when $n\ge 3$.
This group is finitely presentable if $n\ge 4$ \cite{RS} (but not
for $n=3$ \cite{Behr}). In contrast with the previous examples,
the Frobenius morphism $\textnormal{Fr}$ induces an isomorphism
onto a subgroup of {\em infinite} index, and the intersection
$\bigcap_{k\ge 0}\textnormal{Im}(\textnormal{Fr}^k)$ is reduced
to~$\{1\}$.
\end{rem}

Ollivier and Wise also constructed in \cite{OW} the first examples
of non-Hopfian groups with Property~(T). They asked whether a
finitely presented example exists. Although linear finitely
generated groups are residually finite, hence Hopfian, we use them
to answer positively their question.

\begin{thm}
There exists a $S$-arithmetic lattice $\Gamma$, and a central
subgroup $Z\subset \Gamma$, such that $\Gamma$ and $\Gamma/Z$ are
finitely presented, have Property~(T), and $\Gamma/Z$ is
non-Hopfian.\label{t:nonhopf}
\end{thm}

The group $\Gamma$ has a simple description as a matrix group from
which Property~(T) and the non-Hopfian property for $\Gamma/Z$ are
easily checked (Proposition \ref{p:hopfT}). Section \ref{s:3} is
devoted to prove finite presentability of $\Gamma$. We use here a
general criterion for finite presentability of $S$-arithmetic
groups, due to Abels \cite{Abels}. It involves the computation of
the first and second cohomology group of a suitable Lie algebra.


\section{Proofs of all results except finite presentability of~$\Gamma$}

We need some facts about Property~(T).

\begin{lem}[see {\cite[Chap. 3, Th\'eor\`eme 4]{HV}}]
Let $G$ be a locally compact group, and $\Gamma$ a lattice in $G$.
Then $G$ has Property~(T) if and only if $\Gamma$ has
Property~(T).\qed\label{inherited_lattices}
\end{lem}

The next lemma is an immediate consequence of the classification
of semisimple algebraic groups over local fields with Property (T)
(see \cite[Chap.~III, Theorem~5.6]{Margulis}) and S.~P.~Wang's
results on the non-semisimple case \cite[Theorem 2.10]{Wang}.

\begin{lem}
Let $\mathbf{K}$ be a local field, $G$ a connected linear
algebraic group defined over $\mathbf{K}$. Suppose that $G$ is
perfect, and, for every simple quotient $S$ of $G$, either $S$ has
$\mathbf{K}$-rank $\ge 2$, or $\mathbf{K}=\mathbf{R}$ and $S$ is
isogeneous to either $\textnormal{Sp}(n,1)$ ($n\ge 2$) or
$\textnormal{F}_{4(-20)}$. If $\textnormal{char}(\mathbf{K})>0$,
suppose in addition that $G$ has a Levi decomposition defined over
$\mathbf{K}$. Then $G(\mathbf{K})$ has
Property~(T).\qed\label{wang_perfect}
\end{lem}

\begin{proof}[Proof of Proposition \ref{p:Out_infini}] The group
$\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z})$ is linear in dimension $n+m$.
As a semidirect product of two finitely presented groups, it is
finitely presented. For every $k\ge 2$, it is isomorphic to its
proper subgroup $\textnormal{SL}_n(\mathbf{Z})\ltimes
k\textnormal{Mat}_{mn}(\mathbf{Z})$ of finite index $k^{mn}$.

The group $\textnormal{GL}_m(\mathbf{Z})$ acts on
$\textnormal{Mat}_{mn}(\mathbf{Z})$ by right multiplication. Since
this action commutes with the left multiplication of
$\textnormal{SL}_n(\mathbf{Z})$, $\textnormal{GL}_m(\mathbf{Z})$
acts on the semidirect product
$\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z})$ by automorphisms, and, by an
immediate verification, this gives an embedding of
$\textnormal{GL}_m(\mathbf{Z})$ if $n$ is odd or
$\textnormal{PGL}_m(\mathbf{Z})$ if $n$ is even into
$\textnormal{Out}(\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z}))$ (it can be shown that this is
an isomorphism if $n$ is odd; if $n$ is even, the image has index
two). In particular, if $m\ge 2$, then
$\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z})$ has infinite outer automorphism
group.

On the other hand, in view of Lemma \ref{inherited_lattices}, it
has Property~(T) (actually for all $m\ge 0$): indeed,
$\textnormal{SL}_n(\mathbf{Z})\ltimes
\textnormal{Mat}_{mn}(\mathbf{Z})$ is a lattice in
$\textnormal{SL}_n(\mathbf{R})\ltimes
\textnormal{Mat}_{mn}(\mathbf{R})$, which has Property~(T) by
Lemma \ref{wang_perfect} as $n\ge 3$.\end{proof}

We now turn to the proof of Theorem \ref{t:nonhopf}. The following
lemma is immediate, and already used
in \cite{Hall} and~\cite{AbelsPapier}.%REF HALL?

\begin{lem}
Let $\Gamma$ be a group, $Z$ a central subgroup. Let $\alpha$ be
an automorphism of $\Gamma$ such that $\alpha(Z)$ is a proper
subgroup of $Z$. Then $\alpha$ induces a surjective, non-injective
endomorphism of $\Gamma/Z$, whose kernel is
$\alpha^{-1}(Z)/Z$.\qed\label{l:nonhopf}
\end{lem}

\begin{defn}
Fix $n_1,n_2,n_3,n_4\in\mathbf{N}-\{0\}$ with $n_2,n_3\ge 3$. We
set $\Gamma=G(\mathbf{Z}[1/p])$, where $p$ is any prime, and $G$
is algebraic the group defined as matrices by blocks of size
$n_1,n_2,n_3,n_4$:

$$\begin{pmatrix}
  I_{n_1} & (*)_{12} & (*)_{13} & (*)_{14} \\
  0 & (**)_{22} & (*)_{23} & (*)_{24} \\
  0 & 0 & (**)_{33} & (*)_{34} \\
  0 & 0 & 0 & I_{n_4} \\
\end{pmatrix},$$
where $(*)$ denote any matrices and $(**)_{ii}$ denote matrices in
$\textnormal{SL}_{n_i}$, $i=2,3$.

The centre of $G$ consists of matrices of the form
$\begin{pmatrix}
  I_{n_1} & 0 & 0 & (*)_{14} \\
  0 & I_{n_2} & 0 & 0 \\
  0 & 0 & I_{n_3} & 0 \\
  0 & 0 & 0 & I_{n_4} \\
\end{pmatrix}$. Define $Z$
as the centre of $G(\mathbf{Z})$.\label{d:Gamma_nonhopfien}
\end{defn}

\begin{rem}This group is related to an example of Abels: in
\cite{AbelsPapier} he considers the same group, but with blocks
$1\times 1$, and $\textnormal{GL}_1$ instead of
$\textnormal{SL}_1$ in the diagonal. Taking the points over
$\mathbf{Z}[1/p]$, and taking the quotient by a cyclic subgroup if
the centre, this provided the first example of finitely
presentable non-Hopfian solvable group.\end{rem}

\begin{rem}
If we do not care about finite presentability, we can take $n_3=0$
(i.e. 3 blocks suffice).
\end{rem}

We begin by easy observations. Identify $\textnormal{GL}_{n_1}$ to
the upper left diagonal block. It acts by \textit{conjugation} on
$G$ as follows:
$$\begin{pmatrix}
  u & 0 & 0 & 0 \\
  0 & I & 0 & 0 \\
  0 & 0 & I & 0 \\
  0 & 0 & 0 & I \\
\end{pmatrix}\cdot\begin{pmatrix}
  I & A_{12} & A_{13} & A_{14} \\
  0 & B_{2} & A_{23} & A_{24} \\
  0 & 0 & B_3 & A_{34} \\
  0 & 0 & 0 & I \\
\end{pmatrix}=
\begin{pmatrix}
  I & uA_{12} & uA_{13} & uA_{14} \\
  0 & B_{2} & A_{23} & A_{24} \\
  0 & 0 & B_3 & A_{34} \\
  0 & 0 & 0 & I \\
\end{pmatrix}.$$

This gives an action of $\textnormal{GL}_{n_1}$ on $G$, and also
on its centre, and this latter action is faithful. In particular,
for every commutative ring $R$, $\textnormal{GL}_{n_1}(R)$ embeds
in $\textnormal{Out}(G(R))$.

From now on, we suppose that $R=\mathbf{Z}[1/p]$, and
$u=pI_{n_1}$. The automorphism of $\Gamma=G(\mathbf{Z}[1/p])$
induced by $u$ maps $Z$ to its proper subgroup $Z^p$. In view of
Lemma \ref{l:nonhopf}, this implies that $\Gamma/Z$ is
non-Hopfian.

\begin{prop}
The groups $\Gamma$ and $\Gamma/Z$ are finitely generated, have
Property~(T), and $\Gamma/Z$ is non-Hopfian.\label{p:hopfT}
\end{prop}
\begin{proof} We have just proved that $\Gamma/Z$ is non-Hopfian. By the
Borel-Harish-Chandra Theorem \cite{BHC}, $\Gamma$ is a lattice in
$G(\mathbf{R})\times G(\mathbf{Q}_p)$. This group has Property~(T)
as a consequence of Lemma \ref{wang_perfect}. By Lemma
\ref{inherited_lattices}, $\Gamma$ also has Property~(T). Finite
generation is a consequence of Property~(T) \cite[Lemme 10]{HV}.
Since Property~(T) is (trivially) inherited by quotients,
$\Gamma/Z$ also has Property~(T).\end{proof}

\begin{rem}
This group has a surjective endomorphism with nontrivial finite
kernel. We have no analogous example with infinite kernel. Such
examples might be constructed if we could prove that some groups
over rings of dimension $\ge 2$ such as
$\textnormal{SL}_n(\mathbf{Z}[X])$ or
$\textnormal{SL}_n(\mathbf{F}_p[X,Y])$ have Property~(T), but this
is an open problem \cite{Sha}. The non-Hopfian Kazhdan group of
Ollivier and Wise \cite{OW} is torsion-free, so the kernel is
infinite in their case.
\end{rem}

\begin{rem}
It is easy to check that
$\textnormal{GL}_{n_1}(\mathbf{Z})\times\textnormal{GL}_{n_4}(\mathbf{Z})$
embeds in $\textnormal{Out}(\Gamma)$ and
$\textnormal{Out}(\Gamma/Z)$. In particular, if $\max(n_1,n_2)\ge
2$, then these outer automorphism groups are infinite.
\end{rem}

We finish this section by observing that $Z$ is a finitely
generated subgroup of the centre of $\Gamma$, so that finite
presentability of $\Gamma/Z$ immediately follows from that
of~$\Gamma$.

\section{Finite presentability of $\Gamma$}\label{s:3}

We recall that a Hausdorff topological group $H$ is
\textit{compactly presented} if there exists a compact generating
subset $C$ of $H$ such that the abstract group $H$ is the quotient
of the group freely generated by $C$ by relations of bounded
length. See \cite[\S 1.1]{Abels} for more about compact
presentability.

Kneser \cite{Kneser} has proved that for every linear algebraic
$\mathbf{Q}_p$-group, the $S$-arithmetic lattice
$G(\mathbf{Z}[1/p])$ is finitely presented if and only if
$G(\mathbf{Q}_p)$ is compactly presented. A characterization of
the linear algebraic $\mathbf{Q}_p$-groups $G$ such that
$G(\mathbf{Q}_p)$ compactly presented is given in~\cite{Abels}.
This criterion requires the study of a solvable cocompact subgroup
of $G(\mathbf{Q}_p)$, which seems hard to carry out in our
specific example.

Let us describe another sufficient criterion for compact
presentability, also given in \cite{Abels}, which is applicable to
our example. Let $U$ be the unipotent radical in $G$, and let $S$
denote a Levi factor defined over $\mathbf{Q}_p$, so that
$G=S\ltimes U$. Let $\mathfrak{u}$ be the Lie algebra of $U$, and
$D$ be a maximal $\mathbf{Q}_p$-split torus in $S$. We recall that
the first homology group of $\mathfrak{u}$ is defined as the
abelianization
$$H_1(\mathfrak{u})=\mathfrak{u}/[\mathfrak{u},\mathfrak{u}],$$
and the second homology group of $\mathfrak{u}$ is defined as
$\textnormal{Ker}(d_2)/\textnormal{Im}(d_3)$, where the maps
$$\mathfrak{u}\wedge\mathfrak{u}\wedge\mathfrak{u}\stackrel{d_3}
\to\mathfrak{u}\wedge\mathfrak{u}\stackrel{d_2}\to\mathfrak{u}$$
are defined by:

$$d_2(x_1\wedge x_2)=-[x_1,x_2]\;\;\text{and}\;\; d_3(x_1\wedge x_2\wedge
x_3)=x_3 \wedge [x_1,x_2]+x_2\wedge [x_3,x_1]+x_1\wedge
[x_2,x_3].$$


We can now state the result by Abels that we use (see
\cite[Theorem 6.4.3 and Remark 6.4.5]{Abels}).

\begin{thm} Let $G$ be a connected
linear algebraic group over~$\mathbf{Q}_p$. Suppose that the
following assumptions are fulfilled:

\begin{itemize}

\item[(i)] $G$ is $\mathbf{Q}_p$-split.

\item[(ii)] $G$ has no simple quotient of $\mathbf{Q}_p$-rank one.

\item[(iii)] 0 does not lie on the segment joining two dominant
weights for the adjoint representation of $S$ on
$H_1(\mathfrak{u})$.

\item[(iv)] 0 is not a dominant weight for an irreducible
subrepresentation of the adjoint representation of $S$ on
$H_2(\mathfrak{u})$.\end{itemize}

Then $G(\mathbf{Q}_p)$ is compactly presented.\qed\label{t:Abels}
\end{thm}


We now return to our particular example of $G$, observe that it is
clearly $\mathbf{Q}_p$-split, and that its simple quotients are
$\textnormal{SL}_{n_2}$ and $\textnormal{SL}_{n_3}$, which have
rank greater than one. Keep the previous notations $S$, $D$, $U$,
$\mathfrak{u}$, so that $S$ (resp. $D$) denoting in our case the
diagonal by blocks (resp. diagonal) matrices in $G$, and $U$
denotes the matrices in $G$ all of whose diagonal blocks are the
identity. The set of indices of the matrix is partitioned as
$I=I_1\sqcup I_2\sqcup I_3\sqcup I_4$, with $|I_j|=n_j$ as in
Definition \ref{d:Gamma_nonhopfien}. It follows that, for every
field $K$,


$$\mathfrak{u}(K)=\left\{T\in \text{End}(K^I),\;\forall j,\;T(K^{I_j})\subset
\bigoplus_{i<j}K^{I_i}\right\}.$$

Throughout, we use the following notation: a letter such as $i_k$
(or $j_k$, etc.) implicitly means $i_k\in I_k$. Define, in an
obvious way, subgroups $U_{ij}$, $i<j$, of $U$, and their Lie
algebras~$\mathfrak{u}_{ij}$.

%%%%%%%%%%%%%

We begin by checking Condition (iii) of Theorem \ref{t:Abels}.

\begin{lem}
For any two weights of the action of $D$ on $H_1(\mathfrak{u})$, 0
is not on the segment joining them.\label{l:H1_criterion}
\end{lem}
\begin{proof} Recall that $H_1(\mathfrak{u})=\mathfrak{u}/[\mathfrak{u},\mathfrak{u}]$. So it
suffices to look at the action on the supplement $D$-subspace
$\mathfrak{u}_{12}\oplus\mathfrak{u}_{23}\oplus\mathfrak{u}_{34}$
of $[\mathfrak{u},\mathfrak{u}]$. Identifying $S$ with
$\textnormal{SL}_{n_2}\times\textnormal{SL}_{n_3}$, we denote
$(A,B)$ an element of $D\subset S$. We also denote by $e_{pq}$ the
matrix whose coefficient $(p,q)$ equals one and all others are
zero.

$$(A,B)\cdot e_{i_1j_2}=a_{j_2}^{-1}e_{i_1j_2},\quad(A,B)\cdot
e_{j_2k_3}=a_{j_2}b_{k_3}^{-1}e_{j_2k_3},\quad(A,B)\cdot
e_{k_3\ell_4}=b_{k_3}e_{k_3\ell_4}.$$

Since $S=\textnormal{SL}_{n_2}\times \textnormal{SL}_{n_3}$, the
weights for the adjoint action on
$\mathfrak{u}_{12}\oplus\mathfrak{u}_{23}\oplus\mathfrak{u}_{34}$
live in $M/P$, where $M$ is the free $\mathbf{Z}$-module of rank
$n_2+n_3$ with basis $(u_1,\dots,u_{n_2},v_1,\dots,v_{n_3})$, and
$P$ is the plane generated by $\sum_{j_2} u_{j_2}$ and
$\sum_{k_3}v_{k_3}$. Thus, the weights are (modulo $P$)
$-u_{j_2}$, $u_{j_2}-v_{k_3}$, $v_{k_3}$ ($1\le j_2\le n_2$, $1\le
k_3\le n_3$).

Using that $n_2,n_3\ge 3$, it is clear that no nontrivial positive
combination of two weights (viewed as elements of
$\mathbf{Z}^{n_2+n_3}$) lies in $P$.\end{proof}

We must now check Condition (iv) of Theorem \ref{t:Abels}, and
therefore compute $H_2(\mathfrak{u})$ as a $D$-module.


\begin{lem}
$\textnormal{Ker}(d_2)$ is generated by

\begin{itemize}

\item[(1)] $\mathfrak{u}_{12}\wedge \mathfrak{u}_{12}$,
$\mathfrak{u}_{23}\wedge \mathfrak{u}_{23}$,
$\mathfrak{u}_{34}\wedge \mathfrak{u}_{34}$,
$\mathfrak{u}_{13}\wedge\mathfrak{u}_{23}$,
$\mathfrak{u}_{23}\wedge\mathfrak{u}_{24}$,
$\mathfrak{u}_{12}\wedge\mathfrak{u}_{13}$,
$\mathfrak{u}_{24}\wedge\mathfrak{u}_{34}$,
$\mathfrak{u}_{12}\wedge\mathfrak{u}_{34}$.


\item[(2)] $\mathfrak{u}_{14}\wedge\mathfrak{u}$,
$\mathfrak{u}_{13}\wedge \mathfrak{u}_{13}$,
$\mathfrak{u}_{24}\wedge \mathfrak{u}_{24}$,
$\mathfrak{u}_{13}\wedge \mathfrak{u}_{24}$.


\item[(3)] $e_{i_1j_2}\wedge e_{k_2\ell_3}$ ($j_2\neq k_2$),
$e_{i_2j_3}\wedge e_{k_3\ell_4}$ ($j_3\neq \ell_3$).

\item[(4)] $e_{i_1j_2}\wedge e_{k_2\ell_4}$ ($j_2\neq k_2$),
$e_{i_1j_3}\wedge e_{k_3\ell_4}$ ($j_3\neq k_3$).

\item[(5)] Elements of the form
$\sum_{j_2}\alpha_{j_2}(e_{i_1j_2}\wedge e_{j_2k_3})$ if
$\sum_{j_2}\alpha_{j_2}=0$, and
$\sum_{j_3}\alpha_{j_3}(e_{i_2j_3}\wedge e_{j_3k_4})$ if
\linebreak[1]$\sum_{j_3}\alpha_{j_3}=\nolinebreak 0$.

\item[(6)] Elements of the form
$\sum_{j_2}\alpha_{j_2}(e_{i_1j_2}\wedge
e_{j_2k_4})+\sum_{j_3}\beta_{j_3}(e_{i_1j_3}\wedge e_{j_3k_4})$ if
$\sum_{j_2}\alpha_{j_2}+\sum_{j_3}\beta_{j_3}=0$.
\end{itemize}
\label{kerd2}
\end{lem}

\begin{proof} First observe that $\textnormal{Ker}(d_2)$ contains
$\mathfrak{u}_{ij}\wedge\mathfrak{u}_{kl}$ when
$[\mathfrak{u}_{ij},\mathfrak{u}_{kl}]=0$. This corresponds to (1)
and (2). The remaining cases are
$\mathfrak{u}_{12}\wedge\mathfrak{u}_{23}$,
$\mathfrak{u}_{23}\wedge\mathfrak{u}_{34}$,
$\mathfrak{u}_{12}\wedge\mathfrak{u}_{24}$,
$\mathfrak{u}_{13}\wedge\mathfrak{u}_{34}$.

On the one hand, $\textnormal{Ker}(d_2)$ also contains
$e_{i_1j_2}\wedge e_{k_2\ell_3}$ if $j_2\neq k_2$, etc.; this
corresponds to elements in (3), (4). On the other hand,
$d_2(e_{i_1j_2}\wedge e_{j_2k_3})=-e_{i_1k_3}$,
$d_2(e_{i_2j_3}\wedge e_{j_3k_4})=-e_{i_2k_4}$,
$d_2(e_{i_1j_2}\wedge e_{j_2k_4})=-e_{i_1k_4}$,
$d_2(e_{i_1j_3}\wedge e_{j_3k_4})=-e_{i_1k_4}$. The lemma
follows.\end{proof}

\begin{defn}
Denote by $\mathfrak{b}$ (resp. $\mathfrak{h}$) the subspace
generated by elements in (2), (4), and (6) (resp. in (1), (3), and
(5)) of Lemma \ref{kerd2}.
\end{defn}

\begin{prop}
$\textnormal{Im}(d_3)=\mathfrak{b}$, and
$\textnormal{Ker}(d_2)=\mathfrak{b}\oplus\mathfrak{h}$ as
$D$-module. In particular, $H_2(\mathfrak{u})$ is isomorphic to
$\mathfrak{h}$ as a $D$-module.
\end{prop}
\begin{proof} We first prove, in a series of facts, that
$\textnormal{Im}(d_3)\supset\mathfrak{b}$.

\begin{fact}$\mathfrak{u}_{14}\wedge\mathfrak{u}$ is contained in
$\textnormal{Im}(d_3)$.\end{fact}

\begin{proof} If $z\in \mathfrak{u}_{14}$, then $d_3(x\wedge y\wedge z)=z\wedge
[x,y]$. This already shows that $\mathfrak{u}_{14}\wedge
(\mathfrak{u}_{13}\oplus \mathfrak{u}_{24} \oplus
\mathfrak{u}_{14})$ is contained in $\textnormal{Im}(d_3)$, since
$[\mathfrak{u},\mathfrak{u}]=\mathfrak{u}_{13}\oplus\mathfrak{u}_{24}
\oplus \mathfrak{u}_{14}$.

Now, if $(x,y,z)\in
\mathfrak{u}_{24}\times\mathfrak{u}_{12}\times\mathfrak{u}_{34}$,
then $d_3(x\wedge y\wedge z)=z\wedge [x,y]$. Since
$[\mathfrak{u}_{24},\mathfrak{u}_{12}]=\mathfrak{u}_{14}$, this
implies that $\mathfrak{u}_{14}\wedge\mathfrak{u}_{34}\subset
\textnormal{Im}(d_3)$. Similarly,
$\mathfrak{u}_{14}\wedge\mathfrak{u}_{12}\subset
\textnormal{Im}(d_3)$.

Finally we must prove that
$\mathfrak{u}_{14}\wedge\mathfrak{u}_{23}\subset
\textnormal{Im}(d_3)$. This follows from the formula
$e_{i_1j_4}\wedge e_{k_2\ell_3}=d_3(e_{i_1m_2}\wedge
e_{k_2\ell_3}\wedge e_{m_2j_4})$, where $m_2\neq k_2$ (so that we
use that $|I_2|\ge 2$).\end{proof}

\begin{fact}
$\mathfrak{u}_{13}\wedge\mathfrak{u}_{13}$ and, similarly,
$\mathfrak{u}_{24}\wedge\mathfrak{u}_{24}$, are contained in
$\textnormal{Im}(d_3)$.
\end{fact}
\begin{proof} If $(x,y,z)\in
\mathfrak{u}_{12}\times\mathfrak{u}_{23}\times\mathfrak{u}_{13}$,
then $d_3(x\wedge y\wedge z)=z\wedge [x,y]$. Since
$[\mathfrak{u}_{12},\mathfrak{u}_{23}]=\mathfrak{u}_{13}$, this
implies that $\mathfrak{u}_{13}\wedge\mathfrak{u}_{13}\subset
\textnormal{Im}(d_3)$.\end{proof}

\begin{fact}
$\mathfrak{u}_{13}\wedge\mathfrak{u}_{24}$ is contained in
$\textnormal{Im}(d_3)$.
\end{fact}
\begin{proof} $d_3(e_{i_1k_2}\wedge e_{k_2\ell_3}\wedge
e_{k_2j_4})=e_{k_2j_4}\wedge e_{i_1\ell_3}+e_{i_1j_4}\wedge
e_{k_2\ell_3}$. Since we already know that $e_{i_1j_4}\wedge
e_{k_2\ell_3}\in\textnormal{Im}(d_3)$, this implies
$e_{k_2j_4}\wedge
e_{i_1\ell_3}\in\textnormal{Im}(d_3)$.\end{proof}


\begin{fact}
The elements in (4) are in $\textnormal{Im}(d_3)$.
\end{fact}
\begin{proof} $d_3(e_{i_1j_2}\wedge e_{j_2k_3}\wedge e_{\ell_3m_4})=
-e_{i_1k_3}\wedge e_{\ell_3m_4}$ if $k_3\neq\ell_3$. The other
case is similar.\end{proof}

\begin{fact}
The elements in (6) are in $\textnormal{Im}(d_3)$.
\end{fact}
\begin{proof} $d_3(e_{i_1j_2}\wedge e_{j_2k_3}\wedge e_{k_3\ell_4})=
-e_{i_1k_3}\wedge e_{k_3\ell_4}+e_{i_1j_2}\wedge e_{j_2\ell_4}$.
Such elements generate all elements as in (6).\end{proof}


Conversely, we must check
$\textnormal{Im}(d_3)\subset\mathfrak{b}$. By straightforward
verifications:

\begin{itemize}

\item
$d_3(\mathfrak{u}_{14}\wedge\mathfrak{u}\wedge\mathfrak{u})
\subset\mathfrak{u}_{14}\wedge\mathfrak{u}$.

\item
$d_3(\mathfrak{u}_{13}\wedge\mathfrak{u}_{23}\wedge\mathfrak{u}_{24})=0$

\item
$d_3(\mathfrak{u}_{12}\wedge\mathfrak{u}_{13}\wedge\mathfrak{u}_{24})$,
$d_3(\mathfrak{u}_{13}\wedge\mathfrak{u}_{24}\wedge\mathfrak{u}_{34})$,
$d_3(\mathfrak{u}_{12}\wedge\mathfrak{u}_{13}\wedge\mathfrak{u}_{34})$,
$d_3(\mathfrak{u}_{12}\wedge\mathfrak{u}_{24}\wedge\mathfrak{u}_{34})$
are all contained in $\mathfrak{u}_{14}\wedge\mathfrak{u}$.

\item
$d_3(\mathfrak{u}_{12}\wedge\mathfrak{u}_{13}\wedge\mathfrak{u}_{23})\subset
\mathfrak{u}_{13}\wedge\mathfrak{u}_{13}$, and similarly
$d_3(\mathfrak{u}_{23}\wedge\mathfrak{u}_{24}\wedge\mathfrak{u}_{34})\subset
\mathfrak{u}_{24}\wedge\mathfrak{u}_{24}$.

\item
$d_3(\mathfrak{u}_{12}\wedge\mathfrak{u}_{23}\wedge\mathfrak{u}_{24})$
and similarly
$d_3(\mathfrak{u}_{13}\wedge\mathfrak{u}_{23}\wedge\mathfrak{u}_{34})$
are contained in
$\mathfrak{u}_{14}\wedge\mathfrak{u}_{23}+\mathfrak{u}_{13}\wedge\mathfrak{u}_{24}$.

\item The only remaining case is that of
$\mathfrak{u}_{12}\wedge\mathfrak{u}_{23}\wedge\mathfrak{u}_{34}$:
$d_3(e_{i_1j_2}\wedge e_{j'_2k_3}\wedge
e_{k'_3\ell_4})=\delta_{k_3k'_3}e_{i_1j_2}\wedge
e_{j'_2\ell_4}-\delta_{j_2j'_2} e_{i_1k_3}\wedge e_{k'_3\ell_4}$,
which lies in (4) or in (6).

\end{itemize}

Finally $\textnormal{Im}(d_3)=\mathfrak{b}$.

\medskip

It follows from Lemma \ref{kerd2} that
$\textnormal{Ker}(d_2)=\mathfrak{h}\oplus\mathfrak{b}$. Since
$\mathfrak{b}=\textnormal{Im}(d_3)$, this is a $D$-submodule. Let
us check that $\mathfrak{h}$ is also a $D$-submodule; the
computation will be used in the sequel.

The action of $S$ on $\mathfrak{u}$ by {\em conjugation} is given
by:

$$\begin{pmatrix}
  1 & 0 & 0 & 0 \\
  0 & A & 0 & 0 \\
  0 & 0 & B & 0 \\
  0 & 0 & 0 & 1 \\
\end{pmatrix}\cdot\begin{pmatrix}
  0 & X_{12} & X_{13} & X_{14} \\
  0 & 0 & X_{23} & X_{24} \\
  0 & 0 & 0 & X_{34} \\
  0 & 0 & 0 & 0 \\
\end{pmatrix}=\begin{pmatrix}
  0 & X_{12}A^{-1} & X_{13}B^{-1} & X_{14} \\
  0 & 0 & AX_{23}B^{-1} & AX_{24} \\
  0 & 0 & 0 & BX_{34} \\
  0 & 0 & 0 & 0 \\
\end{pmatrix}$$

We must look at the action of $D$ on the elements in (1), (3), and
(5). We fix $(A,B)\in D\subset
S\simeq\textnormal{SL}_{n_2}\times\textnormal{SL}_{n_3}$, and we
write $A=\sum_{j_2}a_{j_2}e_{j_2j_2}$ and
$B=\sum_{k_3}b_{k_3}e_{k_3k_3}$.

\begin{itemize}

\item (1):
\begin{equation}(A,B)\cdot e_{i_1j_2}\wedge
e_{k_1\ell_2}=e_{i_1j_2}A^{-1}\wedge
e_{k_1\ell_2}A^{-1}=a_{j_2}^{-1}a_{\ell_2}^{-1}e_{i_1j_2}\wedge
e_{k_1\ell_2}.\label{e_2}\end{equation} The action on other
elements in (1) has a similar form.

\item (3) ($j_2\neq k_2$):
\begin{equation}(A,B)\cdot e_{i_1j_2}\wedge
e_{k_2\ell_3}=e_{i_1j_2}A^{-1}\wedge
Ae_{k_2\ell_4}B^{-1}=a_{j_2}^{-1}a_{k_2}b_{\ell_3}^{-1}e_{i_1j_2}\wedge
e_{k_2\ell_3}.\label{e_3}\end{equation} The action on the other
elements in (3) has a similar form.

\item (5) ($\sum_{j_2}\alpha_{j_2}=0$)

\begin{align}\nonumber (A,B)\cdot\sum_{j_2}\alpha_{j_2}(e_{i_1j_2}\wedge
e_{j_2k_3})\;\;=\;\;\sum_{j_2}\alpha_{j_2}(e_{i_1j_2}A^{-1}\wedge
Ae_{j_2k_3}B^{-1})\\=\sum_{j_2}\alpha_{j_2}a_{j_2}^{-1}(e_{i_1j_2}\wedge
a_{j_2}b_{k_3}^{-1}e_{j_2k_3})\;\;=\;\;b_{k_3}^{-1}\left(\sum_{j_2}\alpha_{j_2}(e_{i_1j_2}\wedge
e_{j_2k_3})\right).\label{e_5}\end{align}
   The other case in (5) has a similar form.\qedhere\end{itemize}\end{proof}

\begin{lem}
0 is not a weight for the action of $D$ on
$H_2(\mathfrak{u})$.\label{l:H2_criterion}
\end{lem}
\begin{proof} As described in the proof of Lemma \ref{l:H1_criterion}, we
think of weights as elements of $M/P$. Hence, we describe weights
as elements of $M=\mathbf{Z}^{n_2+n_3}$ rather than $M/P$, and
must check that no weight lies in $P$.

\begin{itemize}

\item[(1)] In (\ref{e_2}), the weight is $-u_{j_2}-u_{\ell_2}$,
hence does not belong to $P$ since $n_2\ge 3$. The other
verifications are similar.

\item[(3)] In (\ref{e_3}), the weight is
$-u_{j_2}+u_{k_2}-v_{\ell_3}$, hence does not belong to $P$. The
other verification for (3) is similar.


\item[(5)] In (\ref{e_5}), the weight is $-v_{k_3}$, hence does
dot belong to $P$. The other verification is
similar.\qedhere\end{itemize}
\end{proof}

%%%%%%%%%%%%%%

Finally, Lemmas \ref{l:H1_criterion} and \ref{l:H2_criterion}
imply that the conditions of Theorem \ref{t:Abels} are satisfied,
so that $\Gamma$ is finitely presented.\qed


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bigskip

\noindent {\bf Acknowledgments.} I thank Herbert Abels, Yann
Ollivier, and Fr\'ed\'eric Paulin for useful discussions, and
Laurent Bartholdi and the referee for valuable comments and
corrections.





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% Abstract is required. \begin{abstract} \end{abstract} \maketitle % Text of article.

% Bibliographies can be prepared with BibTeX using amsplain,
% amsalpha, or (for "historical" overviews) natbib style. \bibliographystyle{amsplain}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliographystyle{amsplain}
\begin{thebibliography}{10}


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 {\em An example of a finitely presentable solvable group}.
 In: ``Homological group theory, Proceedings Durham 1977", London Math. Soc. Lecture
Note Ser. {\bf 36} (1979), 105--211.

\bibitem[A2]{Abels} H. {\sc Abels}.
{\em Finite presentability of $S$-arithmetic groups. Compact
presentability of solvable groups}.
 Lecture Notes in Math. {\bf 1261}, Springer, 1987.

\bibitem[Be]{Behr} H. {\sc Behr}.
 {\em Chevalley groups of rank 2 over $\mathbf{F}_q(t)$ are not finitely presentable}.
 In: ``Homological group theory, Proceedings Durham 1977", London Math. Soc. Lecture
Note Ser. {\bf 36} (1979), 213--224.

\bibitem[BHC]{BHC} A. {\sc Borel}, {\sc Harish-Chandra}.
 {\em Arithmetic subgroups of algebraic groups}.
 Ann. of Math. {\bf 75} (1962), 485--535.

%\bibitem[BHV]{BHV} B. {\sc Bekka}, P. {\sc de la Harpe}, Alain {\sc Valette}.
% {\em Kazhdan's Property~(T)}.
% Forthcoming book, 2005, currently available at \url{http://name.math.univ-rennes1.fr/bachir.bekka/}.

\bibitem[BS]{BS} I. {\sc Belegradek}, A. {\sc Szczepa{\'n}ski}.
{\em Endomorphisms of relatively hyperbolic groups}.
 Preprint 2005, arXiv math.GR/0501321.

\bibitem[Ha]{Hall} P. {\sc Hall}.
 {\em The Frattini subgroups of finitely generated groups}.
 Proc. London Math. Soc. (3) {\bf 11} (1961), 327--352.

\bibitem[HV]{HV} P. {\sc de la Harpe}, Alain {\sc Valette}.
 {\em La propri\'et\'e (T) de Kazhdan pour les groupes localement compacts},
 Ast\'erisque {\bf 175}, SMF, 1989.

\bibitem[Kn]{Kneser} M. {\sc Kneser}.
 {\em Erzeugende und Relationen verallgemeinerter Einheitengruppen}.
 J. Reine Angew. Math. {\bf 214/215} (1964), 345--349.

\bibitem[Ma]{Margulis} G. {\sc Margulis}.
{\em Discrete Subgroups of Semisimple Lie Groups}. Springer, 1991.

\bibitem[OW]{OW} Y. {\sc Ollivier}, D. {\sc Wise}.
 {\em Kazhdan groups with infinite outer automorphism group}.
  Preprint 2005, arXiv math.GR/0409203; to appear in Trans. Amer. Math. Soc.

\bibitem[Pa]{Pau} F. {\sc Paulin}.
 {\em Outer automorphisms of hyperbolic groups and small actions on {\bf R}-trees}.
 In ``Arboreal Group Theory'', (MSRI, Berkeley, september 1988)
  R.C. Alperin ed., M.S.R.I. Pub. {\bf 19}, Springer Verlag, 1991.

\bibitem[RS]{RS} U. {\sc Rehrmann}, C. {\sc Soul\'e}.
 {\em Finitely presented groups of matrices}.
 In ``Algebraic K-theory Proceedings Evanston 1976".
 Lecture Notes in Math. {\bf 551}: 164--169, Springer, 1976.

\bibitem[Sh]{Sha} Y. {\sc Shalom}.
 {\em Bounded generation and Kazhdan's property (T)}.
 Publ. Math. Inst. Hautes \'Etudes Sci. {\bf 90} (1999), 145--168.

\bibitem[Wa]{Wang} S. P. {\sc Wang}.
  {\em On the Mautner phenomenon and groups with property (T)}.
 Amer. J. Math. {\bf 104}(6) (1982), 1191--1210.

\bibitem[Wo]{W} Report of the workshop {\em Geometrization of
Kazhdan's Property~(T)} (organizers: B. Bekka, P. de la Harpe, A.
Valette; 2001).
 Unpublished; currently available at
\url{http://www.mfo.de/cgi-bin/tagungsdb?type=21&tnr=0128a}.

\bibitem[Zu]{Zuk} A. {\sc Zuk}.
 {\em La propri\'et\'e (T) de Kazhdan pour les groupes agissant sur les poly\`edres}.
 C. R. Math. Acad. Sci. Paris {\bf 323}, S\'erie I (1996), 453--458.

\end{thebibliography}

%----------------------------------------------------------------------- % End of proc-l.template
%-----------------------------------------------------------------------
\end{document}

math0502140.extracted_bib
\begin{thebibliography}{10}


\bibitem[A1]{AbelsPapier} H. {\sc Abels}.
 {\em An example of a finitely presentable solvable group}.
 In: ``Homological group theory, Proceedings Durham 1977", London Math. Soc. Lecture
Note Ser. {\bf 36} (1979), 105--211.

\bibitem[A2]{Abels} H. {\sc Abels}.
{\em Finite presentability of $S$-arithmetic groups. Compact
presentability of solvable groups}.
 Lecture Notes in Math. {\bf 1261}, Springer, 1987.

\bibitem[Be]{Behr} H. {\sc Behr}.
 {\em Chevalley groups of rank 2 over $\mathbf{F}_q(t)$ are not finitely presentable}.
 In: ``Homological group theory, Proceedings Durham 1977", London Math. Soc. Lecture
Note Ser. {\bf 36} (1979), 213--224.

\bibitem[BHC]{BHC} A. {\sc Borel}, {\sc Harish-Chandra}.
 {\em Arithmetic subgroups of algebraic groups}.
 Ann. of Math. {\bf 75} (1962), 485--535.

%\bibitem[BHV]{BHV} B. {\sc Bekka}, P. {\sc de la Harpe}, Alain {\sc Valette}.
% {\em Kazhdan's Property~(T)}.
% Forthcoming book, 2005, currently available at \url{http://name.math.univ-rennes1.fr/bachir.bekka/}.

\bibitem[BS]{BS} I. {\sc Belegradek}, A. {\sc Szczepa{\'n}ski}.
{\em Endomorphisms of relatively hyperbolic groups}.
 Preprint 2005, arXiv math.GR/0501321.

\bibitem[Ha]{Hall} P. {\sc Hall}.
 {\em The Frattini subgroups of finitely generated groups}.
 Proc. London Math. Soc. (3) {\bf 11} (1961), 327--352.

\bibitem[HV]{HV} P. {\sc de la Harpe}, Alain {\sc Valette}.
 {\em La propri\'et\'e (T) de Kazhdan pour les groupes localement compacts},
 Ast\'erisque {\bf 175}, SMF, 1989.

\bibitem[Kn]{Kneser} M. {\sc Kneser}.
 {\em Erzeugende und Relationen verallgemeinerter Einheitengruppen}.
 J. Reine Angew. Math. {\bf 214/215} (1964), 345--349.

\bibitem[Ma]{Margulis} G. {\sc Margulis}.
{\em Discrete Subgroups of Semisimple Lie Groups}. Springer, 1991.

\bibitem[OW]{OW} Y. {\sc Ollivier}, D. {\sc Wise}.
 {\em Kazhdan groups with infinite outer automorphism group}.
  Preprint 2005, arXiv math.GR/0409203; to appear in Trans. Amer. Math. Soc.

\bibitem[Pa]{Pau} F. {\sc Paulin}.
 {\em Outer automorphisms of hyperbolic groups and small actions on {\bf R}-trees}.
 In ``Arboreal Group Theory'', (MSRI, Berkeley, september 1988)
  R.C. Alperin ed., M.S.R.I. Pub. {\bf 19}, Springer Verlag, 1991.

\bibitem[RS]{RS} U. {\sc Rehrmann}, C. {\sc Soul\'e}.
 {\em Finitely presented groups of matrices}.
 In ``Algebraic K-theory Proceedings Evanston 1976".
 Lecture Notes in Math. {\bf 551}: 164--169, Springer, 1976.

\bibitem[Sh]{Sha} Y. {\sc Shalom}.
 {\em Bounded generation and Kazhdan's property (T)}.
 Publ. Math. Inst. Hautes \'Etudes Sci. {\bf 90} (1999), 145--168.

\bibitem[Wa]{Wang} S. P. {\sc Wang}.
  {\em On the Mautner phenomenon and groups with property (T)}.
 Amer. J. Math. {\bf 104}(6) (1982), 1191--1210.

\bibitem[Wo]{W} Report of the workshop {\em Geometrization of
Kazhdan's Property~(T)} (organizers: B. Bekka, P. de la Harpe, A.
Valette; 2001).
 Unpublished; currently available at
\url{http://www.mfo.de/cgi-bin/tagungsdb?type=21&tnr=0128a}.

\bibitem[Zu]{Zuk} A. {\sc Zuk}.
 {\em La propri\'et\'e (T) de Kazhdan pour les groupes agissant sur les poly\`edres}.
 C. R. Math. Acad. Sci. Paris {\bf 323}, S\'erie I (1996), 453--458.

\end{thebibliography}