First cohomology, rigidity and deformations of isometric group actions
David Fisher
In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group $\G$ to a finite dimensional Lie group $G$ in terms of cohomology of $\G$ with coefficients in the Lie algebra of $G$. This note announces a generalization of Weil's result to a class of homomorphisms into certain infinite dimensional Lie groups, namely diffeomorphism groups of compact manifolds. This gives a criterion for local rigidity of group actions which implies local rigidity of: $(1)$ all isometric actions of groups with property $(T)$, $(2)$ all isometric actions of irreducible lattices in products of simple Lie groups and $(3)$ a certain class of isometric actions of a certain class of cocompact lattices in $SU(1,n)$.
math0411121.tex

\documentclass[11pt]{amsart}
\usepackage{amsmath,amssymb}
\usepackage{setspace,xypic}
%\usepackage[all]{xy}
%\doublespacing
%\pagestyle{empty}
%\oddsidemargin .1cm \evensidemargin .1cm \textwidth 15.8cm
%\oddsidemargin  1in \evensidemargin 1in \textwidth 5.9in
%\topmargin -.15in
%\textheight 9in
%\renewcommand{\thepage}{}


\newcommand\Seqinitial{\xymatrix{
{\Diff^{\infty}(M)}\ar[r]^{P}&{\Diff^{\infty}(M)^k}\ar[r]^{Q}&{\Diff^{\infty}(M)}^r&\\
}}
\newcommand\Seqpoint{\xymatrix{
{\Vect^{\infty}(M)}\ar[r]^{DP_{\Id}}&{\Vect^{\infty}(M)^k}\ar[r]^{DQ_{\pi}}&{\Vect^{\infty}(M)^r}&\\
}}
\newcommand{\A}[3]{{}^{#1} \hskip -2.5pt a^{#2}_{#3}}
\newcommand{\W}[3]{{}^{#1} \hskip -2pt W^{#2}_{#3}}
\newcommand\inG{(G{\times}M)/{\Gamma}}
\newcommand\coin{\alpha:{G{\times}(\inG)}{\rightarrow}{\pi_1(\inG)}}
\newcommand\coGMpi{\alpha:{G{\times}M}{\rightarrow}\pi_1(M)}
\newcommand\colin{{\sigma}{\circ}{\alpha}:{G{\times}(\inG)}{\rightarrow}{GL_n({\mathbb R})}}
\newcommand\rn{{\mathbb R}^n}
\newcommand\base{H/{\Lambda}{\times}M}
\newcommand\cover{H{\times}M}
\newcommand\extension{L}
\newcommand\quotient{Z{\backslash}H}
\newcommand\incon{\theta^{\prime}:{G{\times}(\inG)}{\rightarrow}{\Lambda}}
\newcommand\inv{^{-1}}
\newcommand\fiber{Z{\times}M}
\newcommand\covin{G\times{\tilde M}}
\newcommand\Gco{{\beta}:G{\times}{G/{\Gamma}}{\rightarrow}{\Gamma}}
\newcommand\bundle{\xymatrix{{\cover}\ar[d]_{p}&{Z{\times}M}\ar[l]^{i}\\
{\quotient}\\}}
\newcommand\bundlef{\xymatrix{{\cover}\ar[d]_{f}&{Z{\times}M}\ar[l]^{i}\\
{\quotient}\\}}
\newcommand\inGuc{\xymatrix{{\covin}\ar[d]_{d}&{\Lambda}\ar[l]^{i}\\
{\inG}\\}}
\newcommand\HZ{H_{\mathbb Z}}
\newcommand\LZ{L_{\mathbb Z}}
\newcommand\JZ{J_{\mathbb Z}}
\newcommand\fs{\frak s}
\newcommand\fh{\mathcal H}
\newcommand\fl{\frak l}
\newcommand\fg{\frak g}
\newcommand\fn{\frak n}
\newcommand\fm{\frak m}
\newcommand\ff{\mathfrak F}
\newcommand\fS{\frak S}
\newcommand\fO{\mathcal{O}}
\newcommand\HZS{H_{{\mathbb Z}_S}}
\newcommand\JZS{J_{{\mathbb Z}_S}}
\newcommand\lop{{\Lambda_0}'}
\newcommand\Mp{M^{\prime}}
\newcommand\lo{\Lambda_0}
\newcommand\zl{Z_{\Lambda}}
\newcommand\ul{U_{\Lambda}}
\newcommand\tphi{\tilde {\phi}}
\newcommand\linf{\Lambda_{\infty}}
\newcommand\dinf{\Delta_{\infty}}
\newcommand\LZS{L_{{\mathbb Z}_S}}
\newcommand\UZS{U_{{\mathbb Z}_S}}
\newcommand\uz{\mu_{Z}}
\newcommand\ux{\mu_{X}}
\newcommand\dOp{{\Delta_0}'}
\newcommand\dO{\Delta_0}
\newcommand\NZ{N_{\mathbb Z}}
\newcommand\si{\supp(\mu_i)}
\newcommand\br{\bar \rho}
\newcommand\G{\Gamma}
\newcommand\Lb{\Lambda}
\newcommand\g{\gamma}
\newcommand\Ra{\mathbb R}
\newcommand\Ca{\mathbb C}
\newcommand\Za{\mathbb Z}
\newcommand\Qa{\mathbb Q}
\newcommand\Na{\mathbb N}
\newcommand\Ga{\mathbb G}\newcommand\Ha{\mathbb H}

\DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Isom}{Isom}
\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\im}{im} \DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Diff}{Diff} \DeclareMathOperator{\Rat}{Rat}
\DeclareMathOperator{\Aff}{Aff} \DeclareMathOperator{\Vect}{Vect}
\DeclareMathOperator{\Sect}{Sect}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Homeo}{Homeo}
\DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Exp}{Exp}
\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\CAT}{CAT} \DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Image}{Im}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{defn}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{question}[theorem]{Question}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{tconjecture}[theorem]{Technical Conjecture}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{criterion}[theorem]{Criterion}

\begin{document}

\title[Cohomology and rigidity of isometric actions]{First cohomology, rigidity and deformations of
isometric group actions}
\author{David Fisher}
\thanks{Author partially supported by NSF grants DMS-0226121 and DMS-0541917.}

\begin{abstract}
In 1964, Weil gave a criterion for local rigidity of a
homomorphism from a finitely generated group $\G$ to a finite
dimensional Lie group $G$ in terms of cohomology of $\G$ with
coefficients in the Lie algebra of $G$.  This note announces a
generalization of Weil's result to a class of homomorphisms into
certain infinite dimensional Lie groups, namely diffeomorphism
groups of compact manifolds.  This gives a criterion for local
rigidity of group actions which implies local rigidity of: $(1)$
all isometric actions of groups with property $(T)$, $(2)$ all
isometric actions of irreducible lattices in products of simple
Lie groups and $(3)$ a certain class of isometric actions of a
certain class of cocompact lattices in $SU(1,n)$.
\end{abstract}

\maketitle

\section{A cohomological criterion for local rigidity and applications}
\label{section:results}

In 1964, Andr\'e Weil showed that a homomorphism $\pi$ from a
finitely generated group $\Gamma$ to a Lie group $G$ is {\em
locally rigid} whenever $H^1(\G,\mathfrak g)=0$.  Here $\pi$ is
locally rigid if any nearby homomorphism is conjugate to $\pi$ by
a small element of $G$, $\mathfrak g$ is the Lie algebra of $G$,
and $\G$ acts on $\mathfrak g$ by the composition of $\pi$ and
 the adjoint representation of $G$.  Weil's proof
also applies to $G$ an algebraic group over a local field, but in
all cases, his use of the implicit function theorem forced $G$ to
be finite dimensional.  This note announces the following
generalization of Weil's theorem to some cases where $G$ is an
infinite dimensional Lie group.


\begin{theorem}
\label{theorem:implicit} Let $\Gamma$ be a finitely presented
group, $(M,g)$ a compact Riemannian manifold and
$\pi:\G{\rightarrow}\Isom(M,g){\subset}\Diff^{\infty}(M)$ a
homomorphism. If $H^1(\G,\Vect^{\infty}(M))=0$, the homomorphism
$\pi$ is locally rigid as a homomorphism into $\Diff^{\infty}(M)$.
\end{theorem}

\noindent The proof of Theorem \ref{theorem:implicit} generalizes
Weil's argument, with Hamilton's implicit function theorem in place
of the standard implicit function theorem \cite{Ha,Ha2}. To apply
Hamilton's theorem, I work with a tame exponential map from
$\Vect^{\infty}(M)$ to $\Diff^{\infty}(M)$ which is a local
diffeomorphism. This exponential map is defined by taking a vector
field $V$ to the map $\Exp(V)(x)=\exp_xV_x$ where
$\exp_x:TM_x{\rightarrow}M$ is the Riemannian exponential map. It is
important not to use the map defined by flowing to time one along
$V$, as this has bad geometric properties, e.g. \cite[I.5.5.2]{Ha}.
Much of the proof of Theorem \ref{theorem:implicit} does not require
that the original action preserve a metric, see Theorem
\ref{theorem:affine} below. That result is more difficult than
Weil's, since we need to work with a more delicate implicit function
theorem and since our exponential map is not equivariant. The main
technical difficulty in proving Theorem \ref{theorem:implicit} from
Theorem \ref{theorem:affine} is producing the tame splittings
required by Hamilton's theorem. Tameness is an assumption on Frechet
manifolds and maps between them that is necessary for Hamilton's
Nash-Moser implicit function theorem \cite{Ha,Ha2}. See section
\ref{section:proofsketch} for a more detailed discussion of the
proof. Complete proofs of this result and the applications we now
discuss will appear in \cite{F3}.

One can interpret Theorem \ref{theorem:implicit} more dynamically.
The homomorphism $\pi$ defines an action of $\G$ on $M$ by
isometries and we will abuse notation by using $\pi$ for both the
action and the homomorphism.  Theorem \ref{theorem:implicit} says
that any $C^{\infty}$ action $\pi'$ of $\G$ which is $C^{\infty}$
close to $\pi$ (which is equivalent to a homomorphism
$\pi':\G{\rightarrow}\Diff^{\infty}(M)$ which is close to $\pi$) is
conjugate to $\rho$ by a small $C^{\infty}$ diffeomorphism. In e.g.
\cite{FM2,FM3}, this condition is called {\em $C^{\infty,\infty}$
local rigidity} of $\pi$. I will use this dynamical language when
discussing applications. The question of whether one could prove
local rigidity of a group action by proving vanishing of
$H^1(\G,\Vect(M))$ was first raised in \cite{Z6}, and more
explicitly in the paragraph following the initial statement of
Theorem 5.6 in \cite{Z4}. This question has remained open, but many
authors studied conditions under which $H^1(\G,\Vect(M))$ vanished,
with a variety of assumptions on the regularity of the vector
fields, see for example \cite{H,Ko,L,LZ,Q,Z4}. Vanishing of
$H^1(\G,\Vect(M))$ was labelled {\em infinitesimal rigidity} in the
hope that vanishing of $H^1$ would imply local rigidity. Only the
results in \cite[Section 4]{LZ} apply to isometric actions and so
only those cohomology vanishing theorems yield applications of my
results. In fact, I give a sharper version, Criterion
\ref{criterion:vanishing}, of Theorem \cite[Theorem 4.1]{LZ} in
section \ref{section:applications} and will state below three
applications of Theorem \ref{theorem:implicit} and Criterion
\ref{criterion:vanishing}. In contexts similar to those considered
by the authors listed above, one may be able to verify the
hypothesis of Theorem \ref{theorem:affine} to prove local rigidity
results, see \cite{F3} for further discussion.

In earlier work Benveniste has proven some results on local
rigidity of group actions, also by using Hamilton's implicit
function theorem \cite{Be}. Benveniste's results are special cases
of Theorem \ref{theorem:isomrigid} below. Benveniste uses the
implicit function theorem to show that certain variations of
geometric structures are trivial. These arguments are close in
spirit to Weil's work in \cite{We1,We2} and to the work of Calabi
and Vesentini \cite{C,CV}. My proof is based on a direct analysis
of the space of actions as in Weil's work in \cite{We3}.  Kanai
has also proven local rigidity results by considering variations
of geometric structures, but without using Hamilton's theorem
\cite{Kan}.  For a more detailed history of results on local
rigidity see \cite{F5}.

I will now describe several applications of Theorem
\ref{theorem:isomrigid}, all of which are proven by showing that
$H^1(\G,\Vect^{\infty}(M))=0$.  Using \cite[Theorem 4.1]{LZ}, we
obtain another proof of:

\begin{theorem}\cite{FM2}
\label{theorem:isomrigid} Let $\Gamma$ be a discrete group with
property $(T)$. Then any $C^{\infty}$, Riemannian isometric action
of $\G$ on a compact manifold is $C^{\infty,\infty}$ locally
rigid.
\end{theorem}

Another, more novel, application is:

\begin{theorem}
\label{theorem:irredlattices} Let $\G$ be an irreducible lattice in
a semisimple Lie group with real rank at least two. Then any
$C^{\infty}$, Riemannian isometric action of $\G$ on a compact
manifold is $C^{\infty,\infty}$ locally rigid.
\end{theorem}

\noindent Theorem \ref{theorem:irredlattices} applies to irreducible
lattices in products of rank $1$ groups.  Not all of these groups
have property $(T)$ and so the theorem has many applications not
covered by Theorem \ref{theorem:isomrigid}. For example it applies
to actions of $SL(2,\Za[\sqrt(2)])$. In fact, Theorem
\ref{theorem:irredlattices} applies much more generally than stated.
First, $\G$ can be an irreducible $S$-arithmetic lattice in a
product of simple algebraic groups over different fields. Secondly,
using a result from \cite{Md}, the result can be extended to apply
irreducible lattices in fairly general products of locally compact
topological groups.

For certain cocompact arithmetic lattices $\G$ in a simple group
$G$, the arithmetic structure of $\G$ comes from a realization of
$\G$ as the integer points in $G{\times}K$ where $K$ is a compact
Lie group. In this case it always true that the projection to $G$
is a lattice and the projection to $K$ is dense.  We say a $\G$
action is {\em arithmetic} if it is defined by projecting $\G$ to
$K$, letting $K$ act by $C^{\infty}$ diffeomorphisms on a compact
manifold $M$ and restricting the $K$ action to $\G$. As a
consequence of deep results of Clozel \cite[Theorem 3.2 and
3.5]{Cl1} concerning automorphic forms, Theorem
\ref{theorem:implicit}, and Criterion \ref{criterion:vanishing},
one has the following:

\begin{theorem}
\label{theorem:clozel} For certain principal congruence lattices
$\G<SU(1,n)$, any arithmetic action of $\G$ is $C^{\infty,\infty}$
locally rigid.
\end{theorem}

\noindent Interestingly, some cocompact lattices in $SU(1,n)$ have
homomorphisms $\rho$ to $\Za$ \cite{Ka,BW}, and so have arithmetic
actions with deformations provided the centralizer $Z$ of $K$ in
$\Diff^{\infty}(M)$ is non-trivial.  Having centralizer allows one
to deform the action along the image of the homomorphism
$\rho{\circ}\sigma_t:F{\rightarrow}Z$ where
$\sigma_t:\Za{\rightarrow}Z$ is any one parameter family of
homomorphisms. This construction can also be applied to actions of
lattices in $SO(1,n)$ where having a homomorphism to $\Za$ is much
more common, see e.g. \cite{Lu}. For $\G$ a lattice in $SU(1,n)$ I
know of no example of a faithful $\G$ action with trivial
centralizer which is not locally rigid.

On the other hand for lattices in $SO(1,n)$, even faithful, ergodic
actions with trivial centralizer are not locally rigid. For a {\em
standard lattice} $\G$ constructed using a quadratic form defined
over a quadratic extension of $\Qa$, one has a dense embedding of
$\G$ in $SO(n+1)$.

\begin{theorem}
\label{theorem:so(1,n)} For $\G$ a standard, arithmetic, cocompact
lattice in $SO(1,n)$ as in the last paragraph, and $\rho$ the $\G$
action on $S^n$ defined by the dense embedding of $\G$ in
$SO(n+1)$. Then the space of deformations of $\rho$ is infinite
dimensional.
\end{theorem}

\noindent This construction uses a variant of Johnson-Millson
bending and answers a question in \cite{JM}. It will be discussed in
detail in \cite{F4}.


\section{On the proof of Theorem \ref{theorem:implicit}}
\label{section:proofsketch}

This section outlines the proof of Theorem \ref{theorem:implicit}
without going through the background necessary to define a tame
splitting or to state Hamilton's theorem. In the course of the
outline, I state a more technical result for arbitrary actions,
Theorem \ref{theorem:affine} below.  The end of the section contains
an informal discussion of tameness in the context of Theorem
\ref{theorem:implicit} and an indication of how it is checked.

 Let $M$ be a compact manifold and  $\Vect^{\infty}(M)$ the graded Frechet
space of $C^{\infty}$ vector fields on $M$. Given an affine
connection $\nabla$ there is an $\Aff(M,\nabla)$ equivariant, tame
exponential map $\Exp$ from $\Vect^{\infty}(M)$ to
$C^{\infty}(M,M)$ defined by $\Exp(V)(x)=\exp_x(V_x)$ where
$\exp_x$ is the exponential map defined by $\nabla$. The map
$\Exp$ is a diffeomorphism from a neighborhood $W$ of the zero
vector field to a neighborhood $U$ of the identity in
$\Diff^{\infty}(M){\subset}\Vect^{\infty}(M)$. Note that if
$\nabla$ is not complete, the map $\Exp$ may not be defined on all
of $\Vect^{\infty}(M)$, but this is not relevant to the arguments
given here.

For the remainder of this section, fix a finitely presented group
$\G$ and a presentation of $\G$. This is a finite collection $S$ of
generators $\g_1,{\ldots},\g_k$ and finite collection of relators
$w_1,{\ldots},w_r$ where each $w_i$ is a finite word in the $\g_j$
and their inverses. More formally, we can view each $w_i$ as a word
in an alphabet on $k$ letters.  Let
$\pi:\G{\rightarrow}\Diff^{\infty}(M)$ be a homomorphism, which we
can identify with a point in $\Diff^{\infty}(M)^k$ by taking the
images of the generators. We have a complex:

\begin{equation}
\label{equation:initialsequence} \Seqinitial
\end{equation}

 \noindent Where $P$ is defined by taking
$\psi$ to $(\psi\pi(\g_1)\psi{\inv}, {\ldots},
\psi\pi(\g_k)\psi{\inv})$ and $Q$ is defined by viewing each $r_i$
as a word $w_i$ in $n$ letters and taking
$(\psi_1,{\ldots},\psi_n)$ to
$(w_1(\psi_1,{\ldots},\psi_n),{\ldots},w_k(\psi_1,{\ldots},\psi_n))$.
Since $\Diff^{\infty}(M)$ is proven to be a tame Frechet Lie group
in \cite[Theorem II.2.3.5]{Ha}, it is easy to see that $P$ and $Q$
are smooth and tame.  Letting $\Id$ be the identity map on $M$, it
follows that $P(\Id)=\pi$ and $Q(\pi)=(\Id,{\ldots},\Id)$. Also
note that $Q{\inv}(\Id_M,{\ldots},\Id_M)$ is exactly the space of
$\G$ actions. Note that while this is a closed subset of
$\Diff^{\infty}(M)^k$, it is unclear that it is a manifold even in
a neighborhood of any individual point. (In the finite dimensional
setting, this set is an algebraic variety, and so is a manifold at
``most" points.)

The tangent spaces of $\Diff^{\infty}(M)$ at any point can be
identified with $\Vect^{\infty}(M)$.  To avoid notational
confusion, we let $A=\Diff^{\infty}(M), B=\Diff^{\infty}(M)^k$ and
$C=\Diff^{\infty}(M)^r$.  Then the complex in
$(\ref{equation:initialsequence})$ becomes:

\begin{equation}
\label{equation:letters}
\xymatrix{A\ar[r]^{P}&{B}\ar[r]^{Q}&{C}&\\}
\end{equation}
\noindent and we can also consider the derivative complex of the
complex in $(\ref{equation:letters})$:
\begin{equation}
\label{equation:derivativecomplex}
\xymatrix{TA\ar[r]^{DP}&{TB}\ar[r]^{DQ}&{TC}&\\}
\end{equation}

\noindent It is possible to compute the derivatives $DP_{Id}$ and
$DQ_{\pi}$ explicitly where $\Id$ is the identity map on $M$, and
see that the image of $DP_{\Id}$ is the space of coboundaries
$B^1(\G, \Vect^{\infty}(M))$ and that the kernel of $DQ_{\pi}$ is
the space of cocycles $Z^1(\G,\Vect^{\infty}(M))$. Therefore the
final hypothesis of Theorem \ref{theorem:implicit} means that
$$\frac{\Ker(DQ_{\pi})}{\Image(DP_{\Id})}=0.$$   If
$\Diff^{\infty}(M)$ were finite dimensional, then the implicit
function theorem would immediately imply that in a neighborhood of
$\pi$, the image of $P$ maps onto the set of points in
$\Diff^{\infty}(M)^k$ where $Q$ is equal to $(\Id,{\ldots}\Id)$.
In other words, in a neighborhood of $\pi$, the set of points
which correspond to $\G$ actions is exactly equal to the set of
points in the image of $P$, i.e. the conjugates of $\pi$.  This
would complete the proof and is more or less Weil's proof from
\cite{We3}.

To apply Hamilton's implicit function theorem in this context, we
need to find a tame splitting of equation (\ref{equation:letters}),
and in fact to find a smooth family of tame splittings over a
neighborhood of $U$ of $\Id_M$ in $A$. For a statement of Hamilton's
theorem see \cite{F3} or \cite[Theorem III.1.1.1]{Ha}.  The proof is
\cite[Section 2]{Ha2}. More precisely, we need to find smooth tame
maps $VP$ and $VQ$ that split the sequence


\begin{equation}
\label{equation:splittingsequence} {\xymatrix{
{U{\times}\Vect^{\infty}(M)}\ar[r]^{DP}&{U{\times}\Vect^{\infty}(M)^k }\ar[r]^{DQ}&{U{\times}\Vect^{\infty}(M)^r}&\\
}}
\end{equation}

\noindent i.e. tame smooth maps
$VP:U{\times}\Vect^{\infty}(M)^k{\rightarrow}U{\times}\Vect^{\infty}(M)$
and
$VQ:U{\times}\Vect^{\infty}(M)^r{\rightarrow}U{\times}\Vect^{\infty}(M)^k$
that are each linear in the second variable, such that
$DP{\circ}VP+VQ{\circ}DQ$ is the identity on $\Vect^{\infty}(M)^k$
for any $x{\in}U$.  Note that the maps $P$ and $Q$ are
$\Diff^{\infty}(M)$ equivariant where $\Diff^{\infty}(M)$ acts on
$A$ by translation and on $B$ and $C$ by conjugation. Since the
$\Diff^{\infty}(M)$ action on $A$ is simply transitive, we can use
this to reduce the problem of finding a splitting in a
neighborhood $U$ to the problem of finding a splitting at $\Id$ by
\cite[Lemma 4.3]{Be}. Benveniste's lemma checks that the smooth,
tame splitting at $\Id$, when translated by the
$\Diff^{\infty}(M)$ action, provides a smooth tame splitting on
all of $U$.  It therefore suffices to find a splitting of the
sequence:


\begin{equation}
\label{equation:splittingsequencepoint} \Seqpoint
\end{equation}

\noindent  This yields the following:

\begin{theorem}
\label{theorem:affine} Let $\Gamma$ be a finitely presented group,
$M$ a compact manifold, and $\pi:\G{\rightarrow}\Diff^{\infty}(M)$
a homomorphism. If $H^1(\G,\Vect^{\infty}(M))=0$ and the sequence
in line $(\ref{equation:splittingsequencepoint})$ admits a tame
splitting, then the homomorphism $\pi$ is locally rigid.
\end{theorem}

\noindent See remarks in \cite{F3} for some potential applications
of Theorem \ref{theorem:affine}. In the isometric case, we are
assuming that $H^1(\G,\Vect^{\infty}(M))=0$, so that the map
$DP_{\Id}:{\Image(DP_{\Id})}{\rightarrow}{\Ker(DQ_{\pi})}$ is
surjective. Note that vanishing of $H^1$ does not necessarily imply
that $DP_{\Id}$ has a tame inverse as there is no closed graph
theorem in the tame Frechet category. To find the splitting at $\pi$
and at $Q(\pi)$, we use the fact that $\Vect^{\infty}(M)$ can be
written as a (Hilbertian) direct sum of $\pi(\G)$ invariant finite
dimensional subspaces $V_j$, see the beginning of section
\ref{section:applications} for more discussion. We can then split
the complex in $(\ref{equation:splittingsequence})$ by splitting
each complex in the sequence of complexes:

\begin{equation} \label{equation:splittingfdsequence}
V_j{\longrightarrow}^{\hskip -15pt DP}{\hskip 3pt}V_j^k
{\longrightarrow}^{\hskip -15pt DQ}{\hskip 3pt}V_j^r.
\end{equation}

\noindent Tameness can then be shown using the fact that the $V_j$
are contained in eigenspaces of the Laplacian, the structure of
isometric actions with vanishing cohomology and some estimates on
``small divisors" that yield bounds on the maps used to split the
complex in (\ref{equation:splittingfdsequence}).

\noindent {\bf On tameness:} The formal definition of tameness is
quite involved, so we give an informal discussion in this setting. A
Frechet space is a complete, Hausdorff, locally convex, topological
vector space. The topology on a locally convex, topological vector
space is always defined by a family of semi-norms. The Frechet space
is called {\em graded} when we can choose an increasing family of
semi-norms to define the topology. On $\Vect^{\infty}(M)$, there is
an increasing family of norms, the $C^k$ norms $\|{\cdot}\|_k$ on
vector fields, that define the topology. There is another increasing
family of norms, the $W^{2,k}$ Sobolev norms $\|{\cdot}\|_{2,k}$ on
vector fields. These two families of norms define the same topology
on $\Vect^{\infty}(M)$. More is true, if we normalize the metric so
that $M$ has volume $1$, then the definitions of the norms combined
with the Sobolev embedding theorems imply that:
\begin{equation}
\label{equation:tameequivalence}
\|v\|_{2,k}{\leq}\|v\|_{k}{\leq}C_k\|v\|_{2,k+r}
\end{equation}
for any $v{\in}\Vect^{\infty}(M)$ where $r=\frac{\dim(M)}{2}$ and
$C_k$ is a constant depending on $k$ and the geometry of $M$. This
statement is exactly the statement that these two families of
norms on $\Vect^{\infty}(M)$ are tamely equivalent in the language
of \cite[Definition II.1.1.3]{Ha}.

More generally, given two graded Frechet spaces
$(W,\|{\cdot}\|_j^W)$ and $(V,\|\cdot\|_l^V)$ and a linear map $L$
between them, $L:W{\rightarrow}V$ is tame if there are positive
integers $r$ and $b$ such that:
\begin{equation}
\label{equation:tame} \|Lv\|^V_{j}{\leq}C_j\|v\|^W_{j+r}
\end{equation}
\noindent for every $v{\in}W$ and all $j{\geq}b$ and where $C_j$
is a constant depending on $j$ and $L$. This is \cite[Definition
II.1.2.1]{Ha}.

The complex in line $(\ref{equation:splittingsequence})$ can be
split formally by splitting the complexes in line
$(\ref{equation:splittingfdsequence})$, but we need to see why this
splitting is tame and why the maps constructed preserve smoothness.
Since each $V_j$ is contained in an eigenspace for some eigenvalue
$\lambda_j$ of the Laplacian on vector fields. Letting $I$ be the
identity on $\Vect^{\infty}(M)$, and $\|{\cdot}\|_{2,k}$ the $k$-th
Sobolev norm, we have:

\begin{equation}
\label{equation:normsoneigenspaces}
\|v\|_{2,k}=\|(I+\Delta)^{k/2}v\|_{2,0}=(1+\lambda_j)^{k/2}\|v\|_{2,0}
\end{equation}
\noindent for every $v{\in}V_j$.  The first equality is standard,
and in fact is an equivalent definition for the Sobolev norm, and
the second equality is obvious from the fact that $v{\in}V_j$.

This reduces the problem of splitting the complex in
(\ref{equation:letters}) to the problem of splitting the sequences
in (\ref{equation:splittingfdsequence}) by maps that are bounded by
polynomials in $\lambda_j$.  In the context of Theorem
\ref{theorem:implicit} the bounds are proven using arithmetic
properties of $\pi(\Gamma)$ that we deduce from vanishing of
$H^1(\Gamma, V_j)$ in a finite set of subspaces $V_j$. Tame
estimates on the map $VP$  are proven using the ideas discussed
below in the context of verifying Criterion
\ref{criterion:vanishing}(2). Tame estimates on $VQ$ are much more
involved, see \cite{F3}.

\section{Proving $H^1(\G,\Vect^{\infty}(M))=0$}
\label{section:applications}

For applications of Theorem \ref{theorem:implicit}, I prove a
criterion for the vanishing of $H^1(\G,\Vect^{\infty}(M))$. This
criterion is a sharpening of \cite[Theorem 4.1]{LZ} and the proof is
similar.

 Using that eigenspaces of the
Laplacian on vector fields are finite dimensional and $\Isom(M,g)$
invariant, that $\Isom(M,g)$ is compact and the Peter-Weyl
theorem, we see that the space of vector fields splits as an
infinite direct sum $\oplus_{j=1}^{\infty}V_j$ where each $V_j$ is
a finite dimensional, irreducible $\G$-module, and contained in an
eigenspace for the Laplacian. Let $\lambda_j$ be the eigenvalue
for the eigenspace containing $V_j$.  This is a Hilbertian direct
sum in either the $L^2$ topology or the Sobolev $W^{2,k}$ topology
for any value of $k$. Since each $V_j$ is contained in an
eigenspace for the Laplacian on vector fields, it consists of
$C^{\infty}$ vector fields by a standard elliptic operator
argument.  Fix a finite generating set $S$ for $\Gamma$ and let
$\|{\cdot}\|_2$ denote the $L^2$ norm on vector fields on $M$.

\begin{criterion}
\label{criterion:vanishing} Let $\G$ be a finitely generated
group, $(M,g)$ a compact Riemannian manifold and
$\pi:\G{\rightarrow}\Isom(M,g)$ a homomorphism.  Then the
following are sufficient conditions for vanishing of
$H^1(\G,\Vect^{\infty}(M))$
\begin{enumerate}
\item $H^1(\G,V_j)$ vanishes for every $j$ and \item  there exist
$\varepsilon>0$ and a non-negative integer $\alpha$ such that for
each $V_j$ which is a non-trivial $\G$-module and every
$v_j{\in}V_j$ there is $\g{\in}S$ such that
$$\|v-\pi(\g)v\|_2{\geq}\varepsilon\lambda^{-\alpha}\|v\|_2.$$
\end{enumerate}
\end{criterion}
\noindent Part $(1)$ of the criterion provides the existence of a
formal solution for any cohomological equation coming from
$H^1(\G,\Vect^{\infty}(M))$.  Part $(2)$ guarantees a smooth
solution of a smooth cohomological equation and provides tame
estimates on the size of the solution in terms of the size of the
coefficients in the equation.   Given $v{\in}\Vect^{\infty}(M)$, we
can write $v=\sum_{j}v_j$ and then smoothness is equivalent to
$\sum_j{\lambda_j^n}{\|v_j\|}<\infty$ for all positive integers $n$.
Smoothness of solutions to smooth equations is this fact and
condition $(2)$ in Criterion \ref{criterion:vanishing}.

For all of our applications, Criterion
\ref{criterion:vanishing}$(2)$ can be verified by three methods. The
first is to adapt the argument of the appendix of \cite{Do} to our
more general setting, see also \cite{DoKr} for a related
computation. The second is to use instead a deep arithmetic result
of Clozel, which implies condition $(2)$ with $\alpha=0$,
\cite{Cl1}.  More precisely, Clozel's result implies that all
non-trivial $V_j$ that can arise in this context are outside some
neighborhood of the identity in the Fell topology on the unitary
dual of $\G$.  This uses the fact that all representations of $\G$
which occur, when induced to the group $G$ in which $\G$ is a
lattice, occur in the automorphic spectrum of $G$. For the
generalization of Theorem \ref{theorem:irredlattices} for lattices
in arbitrary products where Clozel's results do not apply, the
reader should observe that any isometric action factors through a
finite group, in which case verifying condition $(2)$ is trivial.
Lastly, we can show, as is done in \cite{F3}, that vanishing of
certain $H^1(\Gamma,V_j)$'s implies certain arithmetic properties of
$\pi(\Gamma)$.  These properties are enough to show that each
$\pi(\gamma)$ satisfies some estimate of diophantine type which is
more than is needed to verify $(2)$.

As remarked above, vanishing of $H^1(\G,\Vect^{\infty}(M))$ in the
context of Theorem \ref{theorem:isomrigid} was already observed in
\cite{LZ}. For Theorem \ref{theorem:irredlattices},  Criterion
\ref{criterion:vanishing}$(1)$ is a consequence of Margulis's
superrigidity theorem \cite[Introduction, Theorem $(3)$]{Ma}. For
the more general lattices discussed in \cite{Md} the vanishing of
cohomology follows from the Monod's criterion for arithmeticity.
More precisely, if the cohomology fails to vanish, Monod's results
imply that $\G$ is $S$-arithmetic and then one can apply Margulis'
result to obtain a contradiction. For Theorem \ref{theorem:clozel},
condition $(1)$ of Criterion \ref{criterion:vanishing} is a special
case of deep results of Clozel \cite[Theorem 3.2 and 3.5]{Cl1}
concerning automorphic forms.


\begin{thebibliography}{Palais}

%\bibitem[Ad]{Ad}
%S. Adams, {\it Some new rigidity results for stable orbit
%equivalence.} Ergodic Theory Dynam. Systems 15 (1995), no. 2,
%209--219.

%\bibitem[AF]{AF}
%Robert A. Adams, John J.F. Fournier, {\it Sobolev Spaces}, 2nd
%edition, Academic Press, 2003.

%\bibitem[A]{A}
%V.I.Arnold, Sur la geometrie differentiale des groups de Lie de dimension infinie et ses
%applications a l'hydrodynamique des fluides parfaits,
%{\it  Ann. Inst. Fourier (Grenoble)} 16 (1966) 319-361.


%\bibitem[B]{B}
%Ballmann, Werner, {\it Lectures on spaces of nonpositive curvature.} With an appendix by Misha Brin. DMV Seminar, 25. Birkh�user Verlag, Basel, 1995.

\bibitem[Be]{Be}
E.J.Benveniste, Rigidity of isometric lattice actions on compact
Riemannian manifolds, {\it GAFA} 10 (2000) 516-542.


%\bibitem[Be2]{Be2}
%E.J.Benveniste, Exotic actions of semisimple groups and their
%deformations, unpublished preprint and  2nd Chapter of University
%of Chicago Ph.d. dissertation, 1996.

%\bibitem[BZ]{BZ}
%I.N.Bernstein and A.V.Zelevinsky, Representations of the group
%$GL(n,F)$ where $F$ is a non-archimedean local field, Russ. Math.
%Surveys, 31.3 (1976), 1-68.

%\bibitem[BF]{BF}
%E.J.Benveniste and D.Fisher, Non-existence of invariant rigid
%structures and invariant almost rigid structures, to appear {\it
%Comm. Anal. Geom.} 21 pages, available at
%http://comet.lehman.cuny.edu/fisher/.

\bibitem[BW]{BW}
Borel, A.; Wallach, N. {\it Continuous cohomology, discrete
subgroups, and representations of reductive groups}. Second edition.
Mathematical Surveys and Monographs, 67. American Mathematical
Society, Providence, RI, 2000.

%\bibitem[BMn]{BMn} Burger, M.; Monod, N. Bounded cohomology of
%lattices in higher rank Lie groups. {\it J. Eur. Math. Soc.
%(JEMS)} 1 (1999), no. 2, 199--235.

%\bibitem[Br]{Br}
%Brunsden, Victor Local rigidity and group cohomology. I. Stowe's
%theorem for Banach manifolds. Bull. Austral. Math. Soc. 59 (1999),
%no. 2, 271--295.

%\bibitem[BMz]{BMz}
%Burger, M.; Mozes, S., ${\CAT}(-1)$-spaces, divergence groups and their commensurators.
%{\it J. Amer. Math. Soc. 9} (1996), no. 1, 57--93.

\bibitem[C]{C}
E. Calabi On compact, Riemannian manifolds with constant curvature.
I. 1961  {\it Proc. Sympos. Pure Math., Vol. III}  pp. 155--180
American Mathematical Society, Providence, R.I.


\bibitem[CV]{CV}
E.Calabi and  E.Vesentini, On compact, locally symmetric K�hler
manifolds. {\it Ann. of Math.} (2) 71 1960 472--507.

%\bibitem[Ca]{Ca}
%J. Cannon. Almost convex groups. {\em Geom. Dedicata}, 22:197-210,
%1987.

%\bibitem[CFGT]{CFGT}
% Cannon, J. W.; Floyd, W. J.; Grayson, M. A.; Thurston, W. P. Solvgroups are not almost convex.
% {em Geom. Dedicata} 31 (1989), no. 3, 291--300.

%\bibitem[Ct]{Ct} Cantat, Serge, Version kaehl\'{e}rienne d'une conjecture de Robert J. Zimmer  ;
% to appear in {\it Annales Scientifiques de l'ENS.}

%S. Cleary and J. Taback. Thompson's group $F$ is not almost
%convex. {\em J. Algebra} 270(1):133-149, 2004


\bibitem[Cl1]{Cl1}
Clozel, Laurent On the cohomology of Kottwitz's arithmetic
varieties. {\it Duke Math. J.} 72 (1993), no. 3, 757--795.

\bibitem[Cl2]{Cl2}
Clozel, Laurent Demonstration de la conjecture $\tau$. (French)
[Proof of the $\tau$-conjecture] {\it Invent. Math.} 151 (2003),
no. 2, 297--328.

%\bibitem[CZ]{CZ}
%K.Corlette and R.J.Zimmer, Superrigidity for cocycles and hyperbolic geometry.
%{\it Internat. J. Math.} 5 (1994) 273--290.

%\bibitem[Co1]{Co1}
%Corlette, Kevin Flat $G$-bundles with canonical metrics. J.
%Differential Geom. 28 (1988), no. 3, 361--382.

%\bibitem[Co2]{Co2} Corlette, Kevin Archimedean superrigidity and hyperbolic
%geometry. Ann. of Math. (2) 135 (1992), no. 1, 165--182.

%\bibitem[CFW]{CFW}
%A. Connes, J. Feldman, and B. Weiss,  {\it An amenable equivalence
%relation is generated by a single transformation.} Ergodic Theory
%Dynamical Systems 1 (1981), no. 4, 431--450 (1982).


%\bibitem[DK]{DK}
%D.Damjanovich and A.Katok, Local Rigidity of Partially Hyperbolic
%Actions on the Torus, preprint, available at
%http://www.math.psu.edu/katok\_a/papers.html.


%\bibitem[De]{De} Delorme, Patrick $1$-cohomologie des repr�sentations unitaires des
%groupes de Lie semi-simples et r�solubles. Produits tensoriels
%continus de repr�sentations. (French){\it Bull. Soc. Math. France}
%105 (1977), no. 3, 281--336.

\bibitem[Do]{Do}
Dolgopyat, Dmitry On mixing properties of compact group extensions
of hyperbolic systems. {\it Israel J. Math.} 130 (2002), 157--205.

\bibitem[DoKr]{DoKr}
D.Dolgopyat and R.Krikorian, On simultaneous linearization of
diffeomorphisms of the sphere. {\it Duke Math. J.} 136 (2007), no.
3, 475--505.

%\bibitem[D]{D}
%Dieudonne, {\it Elements d'Analyse}, vol. 4, Gauthiers-Villars, Paris, 1971.

%\bibitem[EM]{EM}
%D.G.Ebin and J.E.Marden, Groups of diffeomorphisms and the solution of the classical Euler
%equation, {\it Bulliten AMS} 75 (1969) 962-967.

%\bibitem[F1]{F1}
%R. Feres, Connection preserving actions of lattices in $SL_n({\mathbb
%  R})$, {\it Israel J. Math.} 75 (1992), 1-21.

%\bibitem[F1]{F1}
%R. Feres, Affine actions of higher ranks lattices,  {\it
%  Geom. Funct. Ana.} 3 no. 4 (1993), 370-394.

%\bibitem[F2]{F2}
%R. Feres, Actions of discrete linear groups and Zimmer's conjecture,
%{\it J. Diff. Geom.} 42 no. 3 (1995), 554-576.


%\bibitem[Eb]{Eb}
%Eberlein, Patrick. Some properties of the fundamental group of a Fuchsian manifold.
%{\it Invent. Math.} 19 (1973), 5--13.

%\bibitem[E]{E}
%A.Eskin, Quasi-isometric rigidity of nonuniform lattices in higher
%rank symmetric spaces. {\it J. Amer. Math. Soc.} 11 (1998), no. 2,
%321--361.

%\bibitem[EF]{EF}
%Eskin, Alex; Farb, Benson Quasi-flats and rigidity in higher rank
%symmetric spaces. {\it J. Amer. Math. Soc.} 10 (1997), no. 3,
%653--692.

%\bibitem[FH]{FH}
%Faraut, Jacques; Harzallah, Kh�lifa Distances hilbertiennes
%invariantes sur un espace homog�ne. (French) {\it Ann. Inst.
%Fourier (Grenoble)} 24 (1974), no. 3, xiv, 171--217.

%\bibitem[Fa]{Fa}
%Farb, Benson The quasi-isometry classification of lattices in
%semisimple Lie groups. {\it Math. Res. Lett.} 4 (1997), no. 5,
%705--717.

%\bibitem[FS]{FS}
%Farb, Benson; Shalen, Peter Real-analytic actions of lattices.
%{\it Invent. Math.} 135 (1999), no. 2, 273--296.

%\bibitem[FK]{FK}
%R.Feres and A.Katok, Ergodic theory and dynamics of G-spaces. To
%be published as a survey in the volume: {\it Handbook of Dynamical
%Systems}, Elsevier.  Available at:
%http://www.math.wustl.edu/~feres/publications.html.

%\bibitem[F1]{F1} D.Fisher, Deformations of group actions,
%preprint, 14 pages, available at
%http://comet.lehman.cuny.edu/fisher/, submitted Transactions AMS.

%D. Fisher, On the arithmetic structure of lattice actions on compact spaces, to appear
%{\it Erg. Th. Dyn. Sys.}

%\bibitem[F2]{F2}
%D.Fisher, Local linearization at invariant submanifolds for
%actions of groups with property $(T)$, in preparation.

\bibitem[F1]{F3}
D.Fisher First cohomology and local rigidity of group actions,
preprint.

\bibitem[F2]{F4}
D.Fisher, Cohomology of arithmetic groups and bending group actions,
in preparation.

\bibitem[F3]{F5}
D.Fisher, Local rigidity of groups actions: past, present, future,
in {\it Dynamics, Ergodic Theory and Geometry (Mathematical Sciences
Research Institute Publications)}, 45-98, Cambridge University
Press, 2007.

%D. Fisher, Rigid geometric structures and representations of fundamental groups,
%to appear in {\it Rigidity in Geometry and Dynamics}, referreed proceedings, eds.
%M.Burger and A.Iozzi, Springer-Verlag.

%\bibitem[FM1]{FM1}
%D.Fisher and G.A.Margulis, Local rigidity for cocycles,in {\em
%Surv.~Diff.~Geom.~Vol VIII}, refereed volume in honor of Calabi,
%Lawson, Siu and Uhlenbeck , editor: S.T. Yau, 45 pages, 2003.

\bibitem[FM1]{FM2}
D.Fisher and G.A.Margulis, Almost isometric actions, property $(T)$,
and local rigidity.  {\it Invent. Math.}  162  (2005),  no. 1,
19--80.

\bibitem[FM2]{FM3}
D. Fisher and G.A.Margulis, Local rigidity for affine actions of
higher rank Lie groups and their lattices, preprint, 59 pages,
available at http://mypage.iu.edu/~fisherdm/paperwork.html,
submitted.

%\bibitem[HV]{HV}
%P. de la Harpe, A.Valette, {\it La propriete (T) de Kazhdan pour
%les groupes localement compacts}, Asterisque 175, Soc. Math. de
%France, Paris, 1989.

\bibitem[Ha1]{Ha}
Hamilton, Richard S. The inverse function theorem of Nash and
Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65--222.

\bibitem[Ha2]{Ha2}
Hamilton, Richard S. Deformation of complex structures on
manifolds with boundary. I. The stable case. J. Differential
Geometry 12 (1977), no. 1, 1--45.


%\bibitem[HPS]{HPS}
%M.W. Hirsch, C.C. Pugh, and M. Shub, {\it Invariant Manifolds},
%Lecture Notes in Mathematics 583 , Springer-Verlag, New York,
%1977.

\bibitem[H]{H} Hurder, Steven Infinitesimal rigidity for hyperbolic actions.
J. Differential Geom. 41 (1995), no. 3, 515--527.



%\bibitem[Gu]{Gu}
%Guichardet, Alain Sur la cohomologie des groupes topologiques. II.
%(French) {\it Bull. Sci. Math.} (2) 96 (1972), 305--332.

\bibitem[JM]{JM} Johnson, Dennis; Millson, John J. Deformation spaces
associated to compact hyperbolic manifolds. Discrete groups in
geometry and analysis (New Haven, Conn., 1984), 48--106, Progr.
Math., 67, Birkh�user Boston, Boston, MA, 1987.

%\bibitem[JY]{JY}
%J.Jost and S.T.Yau, Harmonic maps and superrigidity, in {\it Lectures on Harmonic Maps} by
%R.Schoen and S.T.Yau, International Press, 1997.



\bibitem[Kan]{Kan}Kanai, M. A new approach to the rigidity of discrete group actions.
{\it Geom. Funct. Anal.} 6 (1996), no. 6, 943--1056.

\bibitem[Ka]{Ka}
Kazhdan, David Some applications of the Weil representation. {\it
J. Analyse Mat.} 32 (1977), 235--248.


\bibitem[Ko]{Ko}
Kononenko, A. Infinitesimal rigidity of boundary lattice actions.
Ergodic Theory Dynam. Systems 19 (1999), no. 1, 35--60.

\bibitem[L]{L}
J. Lewis, Infinitesimal rigidity for the action of $SL_n({\mathbb
Z})$ on ${\mathbb T}^n$, {\it Trans. Amer. Math. Soc.} 324 no. 1
(1991) 421-445.

\bibitem[LZ]{LZ}
Lubotzky, Alexander; Zimmer, Robert J. Variants of Kazhdan's
property for subgroups of semisimple groups. {\it Israel J. Math.}
66 (1989), no. 1-3, 289--299.


\bibitem[Lu]{Lu}
Lubotzky, Alexander; Eigenvalues of the Laplacian, the first Betti
number and the congruence subgroup problem. {\it Ann. of Math.}
144 (1996), no. 2, 441--452.



\bibitem[Ma]{Ma}
G.A.Margulis, {\it Discrete subgroups of semisimple Lie groups},
Springer-Verlag, New York, 1991.

%\bibitem[Md1]{Md1}
%Monod, Nicolas, Superrigidity for irreducible lattices and
%geometric splitting, preprint.


\bibitem[Md]{Md}
Monod, Nicolas Arithmeticity vs. nonlinearity for irreducible
lattices.  Geom. Dedicata  112  (2005), 225--237.

\bibitem[Q]{Q} Qian, Nantian Infinitesimal rigidity of higher rank lattice
actions. Comm. Anal. Geom. 4 (1996), no. 3, 495--524.

%\bibitem[Rg]{Rg}
%M.S. Raghunathan, On the first cohomology of discrete subgroups of
%semisimple Lie groups. {\it Amer. J. Math.} 87 (1965) 103--139.




\bibitem[We1]{We1}Weil, Andr\'e On discrete subgroups of Lie groups.
{\it Ann. of Math.}  72 (1960) 369--384.

\bibitem[We2]{We2}
Weil, Andr\'e On discrete subgroups of Lie groups II. {\it Ann. of
Math.}  75 (1962) 578--602.

\bibitem[We3]{We3}
 Weil, Andr\'e Remarks on the cohomology of groups.
 {\it Ann. of Math.}  80 (1964) 149--157.

\bibitem[Z1]{Z6}
R.J. Zimmer, Actions of semisimple groups and discrete subgroups,
{\it Proc. Internat. Cong. Math.}, Berkeley, 1986, 1247-1258.

\bibitem[Z2]{Z4}
Zimmer, Robert J. Infinitesimal rigidity for smooth actions of
discrete subgroups of Lie groups. J. Differential Geom. 31 (1990),
no. 2, 301--322.



\end{thebibliography}
\bigskip
\noindent Department of Mathematics, Indiana University,
Bloomington, IN 47401\\




\end{document}

math0411121.extracted_bib
\begin{thebibliography}{Palais}

%\bibitem[Ad]{Ad}
%S. Adams, {\it Some new rigidity results for stable orbit
%equivalence.} Ergodic Theory Dynam. Systems 15 (1995), no. 2,
%209--219.

%\bibitem[AF]{AF}
%Robert A. Adams, John J.F. Fournier, {\it Sobolev Spaces}, 2nd
%edition, Academic Press, 2003.

%\bibitem[A]{A}
%V.I.Arnold, Sur la geometrie differentiale des groups de Lie de dimension infinie et ses
%applications a l'hydrodynamique des fluides parfaits,
%{\it  Ann. Inst. Fourier (Grenoble)} 16 (1966) 319-361.


%\bibitem[B]{B}
%Ballmann, Werner, {\it Lectures on spaces of nonpositive curvature.} With an appendix by Misha Brin. DMV Seminar, 25. Birkh�user Verlag, Basel, 1995.

\bibitem[Be]{Be}
E.J.Benveniste, Rigidity of isometric lattice actions on compact
Riemannian manifolds, {\it GAFA} 10 (2000) 516-542.


%\bibitem[Be2]{Be2}
%E.J.Benveniste, Exotic actions of semisimple groups and their
%deformations, unpublished preprint and  2nd Chapter of University
%of Chicago Ph.d. dissertation, 1996.

%\bibitem[BZ]{BZ}
%I.N.Bernstein and A.V.Zelevinsky, Representations of the group
%$GL(n,F)$ where $F$ is a non-archimedean local field, Russ. Math.
%Surveys, 31.3 (1976), 1-68.

%\bibitem[BF]{BF}
%E.J.Benveniste and D.Fisher, Non-existence of invariant rigid
%structures and invariant almost rigid structures, to appear {\it
%Comm. Anal. Geom.} 21 pages, available at
%http://comet.lehman.cuny.edu/fisher/.

\bibitem[BW]{BW}
Borel, A.; Wallach, N. {\it Continuous cohomology, discrete
subgroups, and representations of reductive groups}. Second edition.
Mathematical Surveys and Monographs, 67. American Mathematical
Society, Providence, RI, 2000.

%\bibitem[BMn]{BMn} Burger, M.; Monod, N. Bounded cohomology of
%lattices in higher rank Lie groups. {\it J. Eur. Math. Soc.
%(JEMS)} 1 (1999), no. 2, 199--235.

%\bibitem[Br]{Br}
%Brunsden, Victor Local rigidity and group cohomology. I. Stowe's
%theorem for Banach manifolds. Bull. Austral. Math. Soc. 59 (1999),
%no. 2, 271--295.

%\bibitem[BMz]{BMz}
%Burger, M.; Mozes, S., ${\CAT}(-1)$-spaces, divergence groups and their commensurators.
%{\it J. Amer. Math. Soc. 9} (1996), no. 1, 57--93.

\bibitem[C]{C}
E. Calabi On compact, Riemannian manifolds with constant curvature.
I. 1961  {\it Proc. Sympos. Pure Math., Vol. III}  pp. 155--180
American Mathematical Society, Providence, R.I.


\bibitem[CV]{CV}
E.Calabi and  E.Vesentini, On compact, locally symmetric K�hler
manifolds. {\it Ann. of Math.} (2) 71 1960 472--507.

%\bibitem[Ca]{Ca}
%J. Cannon. Almost convex groups. {\em Geom. Dedicata}, 22:197-210,
%1987.

%\bibitem[CFGT]{CFGT}
% Cannon, J. W.; Floyd, W. J.; Grayson, M. A.; Thurston, W. P. Solvgroups are not almost convex.
% {em Geom. Dedicata} 31 (1989), no. 3, 291--300.

%\bibitem[Ct]{Ct} Cantat, Serge, Version kaehl\'{e}rienne d'une conjecture de Robert J. Zimmer  ;
% to appear in {\it Annales Scientifiques de l'ENS.}

%S. Cleary and J. Taback. Thompson's group $F$ is not almost
%convex. {\em J. Algebra} 270(1):133-149, 2004


\bibitem[Cl1]{Cl1}
Clozel, Laurent On the cohomology of Kottwitz's arithmetic
varieties. {\it Duke Math. J.} 72 (1993), no. 3, 757--795.

\bibitem[Cl2]{Cl2}
Clozel, Laurent Demonstration de la conjecture $\tau$. (French)
[Proof of the $\tau$-conjecture] {\it Invent. Math.} 151 (2003),
no. 2, 297--328.

%\bibitem[CZ]{CZ}
%K.Corlette and R.J.Zimmer, Superrigidity for cocycles and hyperbolic geometry.
%{\it Internat. J. Math.} 5 (1994) 273--290.

%\bibitem[Co1]{Co1}
%Corlette, Kevin Flat $G$-bundles with canonical metrics. J.
%Differential Geom. 28 (1988), no. 3, 361--382.

%\bibitem[Co2]{Co2} Corlette, Kevin Archimedean superrigidity and hyperbolic
%geometry. Ann. of Math. (2) 135 (1992), no. 1, 165--182.

%\bibitem[CFW]{CFW}
%A. Connes, J. Feldman, and B. Weiss,  {\it An amenable equivalence
%relation is generated by a single transformation.} Ergodic Theory
%Dynamical Systems 1 (1981), no. 4, 431--450 (1982).


%\bibitem[DK]{DK}
%D.Damjanovich and A.Katok, Local Rigidity of Partially Hyperbolic
%Actions on the Torus, preprint, available at
%http://www.math.psu.edu/katok\_a/papers.html.


%\bibitem[De]{De} Delorme, Patrick $1$-cohomologie des repr�sentations unitaires des
%groupes de Lie semi-simples et r�solubles. Produits tensoriels
%continus de repr�sentations. (French){\it Bull. Soc. Math. France}
%105 (1977), no. 3, 281--336.

\bibitem[Do]{Do}
Dolgopyat, Dmitry On mixing properties of compact group extensions
of hyperbolic systems. {\it Israel J. Math.} 130 (2002), 157--205.

\bibitem[DoKr]{DoKr}
D.Dolgopyat and R.Krikorian, On simultaneous linearization of
diffeomorphisms of the sphere. {\it Duke Math. J.} 136 (2007), no.
3, 475--505.

%\bibitem[D]{D}
%Dieudonne, {\it Elements d'Analyse}, vol. 4, Gauthiers-Villars, Paris, 1971.

%\bibitem[EM]{EM}
%D.G.Ebin and J.E.Marden, Groups of diffeomorphisms and the solution of the classical Euler
%equation, {\it Bulliten AMS} 75 (1969) 962-967.

%\bibitem[F1]{F1}
%R. Feres, Connection preserving actions of lattices in $SL_n({\mathbb
%  R})$, {\it Israel J. Math.} 75 (1992), 1-21.

%\bibitem[F1]{F1}
%R. Feres, Affine actions of higher ranks lattices,  {\it
%  Geom. Funct. Ana.} 3 no. 4 (1993), 370-394.

%\bibitem[F2]{F2}
%R. Feres, Actions of discrete linear groups and Zimmer's conjecture,
%{\it J. Diff. Geom.} 42 no. 3 (1995), 554-576.


%\bibitem[Eb]{Eb}
%Eberlein, Patrick. Some properties of the fundamental group of a Fuchsian manifold.
%{\it Invent. Math.} 19 (1973), 5--13.

%\bibitem[E]{E}
%A.Eskin, Quasi-isometric rigidity of nonuniform lattices in higher
%rank symmetric spaces. {\it J. Amer. Math. Soc.} 11 (1998), no. 2,
%321--361.

%\bibitem[EF]{EF}
%Eskin, Alex; Farb, Benson Quasi-flats and rigidity in higher rank
%symmetric spaces. {\it J. Amer. Math. Soc.} 10 (1997), no. 3,
%653--692.

%\bibitem[FH]{FH}
%Faraut, Jacques; Harzallah, Kh�lifa Distances hilbertiennes
%invariantes sur un espace homog�ne. (French) {\it Ann. Inst.
%Fourier (Grenoble)} 24 (1974), no. 3, xiv, 171--217.

%\bibitem[Fa]{Fa}
%Farb, Benson The quasi-isometry classification of lattices in
%semisimple Lie groups. {\it Math. Res. Lett.} 4 (1997), no. 5,
%705--717.

%\bibitem[FS]{FS}
%Farb, Benson; Shalen, Peter Real-analytic actions of lattices.
%{\it Invent. Math.} 135 (1999), no. 2, 273--296.

%\bibitem[FK]{FK}
%R.Feres and A.Katok, Ergodic theory and dynamics of G-spaces. To
%be published as a survey in the volume: {\it Handbook of Dynamical
%Systems}, Elsevier.  Available at:
%http://www.math.wustl.edu/~feres/publications.html.

%\bibitem[F1]{F1} D.Fisher, Deformations of group actions,
%preprint, 14 pages, available at
%http://comet.lehman.cuny.edu/fisher/, submitted Transactions AMS.

%D. Fisher, On the arithmetic structure of lattice actions on compact spaces, to appear
%{\it Erg. Th. Dyn. Sys.}

%\bibitem[F2]{F2}
%D.Fisher, Local linearization at invariant submanifolds for
%actions of groups with property $(T)$, in preparation.

\bibitem[F1]{F3}
D.Fisher First cohomology and local rigidity of group actions,
preprint.

\bibitem[F2]{F4}
D.Fisher, Cohomology of arithmetic groups and bending group actions,
in preparation.

\bibitem[F3]{F5}
D.Fisher, Local rigidity of groups actions: past, present, future,
in {\it Dynamics, Ergodic Theory and Geometry (Mathematical Sciences
Research Institute Publications)}, 45-98, Cambridge University
Press, 2007.

%D. Fisher, Rigid geometric structures and representations of fundamental groups,
%to appear in {\it Rigidity in Geometry and Dynamics}, referreed proceedings, eds.
%M.Burger and A.Iozzi, Springer-Verlag.

%\bibitem[FM1]{FM1}
%D.Fisher and G.A.Margulis, Local rigidity for cocycles,in {\em
%Surv.~Diff.~Geom.~Vol VIII}, refereed volume in honor of Calabi,
%Lawson, Siu and Uhlenbeck , editor: S.T. Yau, 45 pages, 2003.

\bibitem[FM1]{FM2}
D.Fisher and G.A.Margulis, Almost isometric actions, property $(T)$,
and local rigidity.  {\it Invent. Math.}  162  (2005),  no. 1,
19--80.

\bibitem[FM2]{FM3}
D. Fisher and G.A.Margulis, Local rigidity for affine actions of
higher rank Lie groups and their lattices, preprint, 59 pages,
available at http://mypage.iu.edu/~fisherdm/paperwork.html,
submitted.

%\bibitem[HV]{HV}
%P. de la Harpe, A.Valette, {\it La propriete (T) de Kazhdan pour
%les groupes localement compacts}, Asterisque 175, Soc. Math. de
%France, Paris, 1989.

\bibitem[Ha1]{Ha}
Hamilton, Richard S. The inverse function theorem of Nash and
Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65--222.

\bibitem[Ha2]{Ha2}
Hamilton, Richard S. Deformation of complex structures on
manifolds with boundary. I. The stable case. J. Differential
Geometry 12 (1977), no. 1, 1--45.


%\bibitem[HPS]{HPS}
%M.W. Hirsch, C.C. Pugh, and M. Shub, {\it Invariant Manifolds},
%Lecture Notes in Mathematics 583 , Springer-Verlag, New York,
%1977.

\bibitem[H]{H} Hurder, Steven Infinitesimal rigidity for hyperbolic actions.
J. Differential Geom. 41 (1995), no. 3, 515--527.



%\bibitem[Gu]{Gu}
%Guichardet, Alain Sur la cohomologie des groupes topologiques. II.
%(French) {\it Bull. Sci. Math.} (2) 96 (1972), 305--332.

\bibitem[JM]{JM} Johnson, Dennis; Millson, John J. Deformation spaces
associated to compact hyperbolic manifolds. Discrete groups in
geometry and analysis (New Haven, Conn., 1984), 48--106, Progr.
Math., 67, Birkh�user Boston, Boston, MA, 1987.

%\bibitem[JY]{JY}
%J.Jost and S.T.Yau, Harmonic maps and superrigidity, in {\it Lectures on Harmonic Maps} by
%R.Schoen and S.T.Yau, International Press, 1997.



\bibitem[Kan]{Kan}Kanai, M. A new approach to the rigidity of discrete group actions.
{\it Geom. Funct. Anal.} 6 (1996), no. 6, 943--1056.

\bibitem[Ka]{Ka}
Kazhdan, David Some applications of the Weil representation. {\it
J. Analyse Mat.} 32 (1977), 235--248.


\bibitem[Ko]{Ko}
Kononenko, A. Infinitesimal rigidity of boundary lattice actions.
Ergodic Theory Dynam. Systems 19 (1999), no. 1, 35--60.

\bibitem[L]{L}
J. Lewis, Infinitesimal rigidity for the action of $SL_n({\mathbb
Z})$ on ${\mathbb T}^n$, {\it Trans. Amer. Math. Soc.} 324 no. 1
(1991) 421-445.

\bibitem[LZ]{LZ}
Lubotzky, Alexander; Zimmer, Robert J. Variants of Kazhdan's
property for subgroups of semisimple groups. {\it Israel J. Math.}
66 (1989), no. 1-3, 289--299.


\bibitem[Lu]{Lu}
Lubotzky, Alexander; Eigenvalues of the Laplacian, the first Betti
number and the congruence subgroup problem. {\it Ann. of Math.}
144 (1996), no. 2, 441--452.



\bibitem[Ma]{Ma}
G.A.Margulis, {\it Discrete subgroups of semisimple Lie groups},
Springer-Verlag, New York, 1991.

%\bibitem[Md1]{Md1}
%Monod, Nicolas, Superrigidity for irreducible lattices and
%geometric splitting, preprint.


\bibitem[Md]{Md}
Monod, Nicolas Arithmeticity vs. nonlinearity for irreducible
lattices.  Geom. Dedicata  112  (2005), 225--237.

\bibitem[Q]{Q} Qian, Nantian Infinitesimal rigidity of higher rank lattice
actions. Comm. Anal. Geom. 4 (1996), no. 3, 495--524.

%\bibitem[Rg]{Rg}
%M.S. Raghunathan, On the first cohomology of discrete subgroups of
%semisimple Lie groups. {\it Amer. J. Math.} 87 (1965) 103--139.




\bibitem[We1]{We1}Weil, Andr\'e On discrete subgroups of Lie groups.
{\it Ann. of Math.}  72 (1960) 369--384.

\bibitem[We2]{We2}
Weil, Andr\'e On discrete subgroups of Lie groups II. {\it Ann. of
Math.}  75 (1962) 578--602.

\bibitem[We3]{We3}
 Weil, Andr\'e Remarks on the cohomology of groups.
 {\it Ann. of Math.}  80 (1964) 149--157.

\bibitem[Z1]{Z6}
R.J. Zimmer, Actions of semisimple groups and discrete subgroups,
{\it Proc. Internat. Cong. Math.}, Berkeley, 1986, 1247-1258.

\bibitem[Z2]{Z4}
Zimmer, Robert J. Infinitesimal rigidity for smooth actions of
discrete subgroups of Lie groups. J. Differential Geom. 31 (1990),
no. 2, 301--322.



\end{thebibliography}