First cohomology and local rigidity\\ of group actions
David Fisher
Given a topological group $G$, and a finitely generated group $\G$, a homomorphism $\pi:\G{\rightarrow}G$ is {\em locally rigid} if any nearby by homomorphism $\pi'$ is conjugate to $\pi$ by a small element of $G$. 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 $G$. Here we generalize Weil's result to a class of homomorphisms into certain infinite dimensional Lie groups, namely groups of diffeomorphism 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 certain more general locally compact groups and $(3)$ a certain class of isometric actions of a certain class of cocompact lattices in $SU(1,n)$.
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\begin{document}
%\currannalsline{0}{2006}

\title{First cohomology and local rigidity\\ of group actions}


\acknowledgements{Author partially supported by NSF grants
DMS-0226121 and DMS-0541917.}
\author{David Fisher}
\institution{Department of Mathematics, Indiana University, Bloomington, IN 47405\\
\email{fisherdm@indiana.edu}}

\begin{abstract}
Given a topological group $G$, and a finitely generated group
$\G$, a homomorphism $\pi:\G{\rightarrow}G$ is {\em locally rigid}
if any nearby by homomorphism $\pi'$ is conjugate to $\pi$ by a
small element of $G$. 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 $G$. Here we generalize
Weil's result to a class of homomorphisms into certain infinite
dimensional Lie groups, namely groups of diffeomorphism 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
certain more general locally compact 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 of
characteristic zero, but his use of the implicit function theorem
forced $G$ to be finite dimensional.  Here we prove 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{\bf Remarks:}\begin{enumerate} \item We also state a
variant of Theorem \ref{theorem:implicit} where $\pi(\G)$ is not
assumed to preserve a metric.  This result, Theorem
\ref{theorem:affine}, gives a completely general condition for
local ridigity of  $\pi:\G{\rightarrow}\Diff^{\infty}(M)$ in terms
of ``effective vanishing" of $H^1(\G,\Vect^{\infty}(M))$. As the
paper \cite{H1} led to a long series of local rigidity theorems
proved using techniques of hyperbolic dynamics, this paper, and
particularly Theorem \ref{theorem:affine}, open the way for a
wide-ranging geometric approach to questions of local rigidity.
See the end of subsection \ref{subsection:affine} as well as
\cite{F2} for discussion.


\item Variants of the theorem where $\Diff^{\infty}(M)$ is
replaced by a tame Frechet subgroup are also true and follow from
the same proof.  See subsection \ref{subsection:subgroups} for
discussion.

\item The $\Gamma$ action on $\Vect^{\infty}(M)$ will be denoted
by $d\pi$ and
$(d\pi(\gamma)V)(x)=D\pi(\gamma)_{\pi(\gamma){\inv}x}V(\pi(\gamma){\inv}x)$
where $Df$ is the differential of $f$.  That this yields a
$\Gamma$ action is simply the chain rule.

%\item Theorem \ref{theorem:implicit} applies to the trivial action
%of any group on any manifold, in which case local rigidity is
%equivalent to $H^1(\G,\Ra)=0$. This result for the trivial action
%as well as a more general technical result follow from work of
%Fleming \cite{Fl}. Fleming's theorem's can be combined with work
%of Mok \cite{Mk} to prove (very) special cases of Theorem
%\ref{theorem:isomrigid}. See \cite{F2} for further discussion.
\end{enumerate}

\noindent One can rephrase Theorem \ref{theorem:implicit}
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 $\pi$ 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 discuss applications in this dynamical
language. 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}, see also \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. The results in \cite[Section 4]{LZ}
apply to isometric actions and so that cohomology vanishing theorem
yields an application of Theorem \ref{theorem:implicit}. As remarked
above Theorem \ref{theorem:affine} below should have applications in
contexts similar to those of \cite{H,Ko,L,Q,Z4}, but these
applications require a stronger condition than vanishing of
$H^1(\G,\Vect^{\infty}(M))$.

To give applications of Theorem \ref{theorem:implicit}, in section
\ref{section:applications}, I establish Criterion
\ref{criterion:vanishing}, which strengthens \cite[Theorem
4.1]{LZ}. I will now describe three applications of Theorem
\ref{theorem:implicit}, all of which are proven using Criterion
\ref{criterion:vanishing} to show vanishing of
$H^1(\Vect^{\infty}(M))$.  Criterion \ref{criterion:vanishing} can
be verified easily for many actions using results on vanishing of
finite dimensional cohomology groups in combination with
elementary results on equidistribution.   Using Criterion
\ref{criterion:vanishing} or \cite[Theorem 4.1]{LZ}, we obtain a
more geometric proof of the following:

\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}

\noindent Another, more novel, application is:

\begin{theorem}
\label{theorem:irredlattices} Let $\G$ be an irreducible lattice
in a semisimple Lie group $G$ 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 this theorem has many applications not
covered by Theorem \ref{theorem:isomrigid}. The simplest new group
to whose actions this theorem applies is $SL(2,\Za[\sqrt{2}])$ and
one can construct many interesting examples of groups and actions
not covered by previous work. 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. Also, using a
result from \cite{Md}, the result can be extended to apply to
irreducible lattices in fairly general products of locally compact
topological groups.  By results in \cite{Md}, unless $\G$ is
actually $S$-arithmetic,  all isometric $\G$ actions on compact
manifolds factor through a finite quotient of $\G$, but there is no
a priori reason that such isometric actions are not ``close to"
faithful, non-isometric actions.  For precise statements of these
variants of Theorem \ref{theorem:irredlattices}, see subsection
\ref{subsection:irredlattices}.

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 $\G$. Using 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:

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

\noindent Here by {\em congruence lattice}, we mean an arithmetic
lattice which contains a congruence subgroup as a subgroup of finite
index.  The class of lattices to which Theorem \ref{theorem:clozel}
applies are called fundamental groups of Kottwitz varieties in
\cite{Cl1} and are described below in subsection
\ref{subsection:clozel}.  Interestingly, some cocompact congruence
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 isometric $\G$ action with trivial
centralizer which is not locally rigid.  In general the question of
when cohomology classes in $H^1(\G,\Vect(M))$ integrate to
deformations is quite difficult, see \cite{F4} for a discussion of
some examples.  In particular, that paper describes isometric
actions of lattices in $SO(1,n)$ on $S^n$ which have no centralizer
and admit many deformations.

The proof of Theorem \ref{theorem:implicit}, while modelled on
Weil's proof of his results in \cite{We3}, is significantly more
difficult.  The key step in Weil's proof is an application of the
implicit function theorem.  Our work requires that we use the more
difficult and delicate implicit function theorem of Hamilton in
place of the standard implicit function theorem \cite{Ha,Ha2}. To
apply Hamilton's theorem, I work with a tame, locally surjective,
exponential map from $\Vect^{\infty}(M)$ to $\Diff^{\infty}(M)$.
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
exponential map for some affine connection on $M$. 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}.  Another step
in following Weil's argument that is also substantially more
difficult in this setting is the computation relating the formal
output of the implicit function theorem, recorded here as
Proposition \ref{proposition:fromhamilton}, to cohomology of $\G$ in
some module. In Weil's setting, this follows almost trivially from
equivariance of the exponential map. In order to use Hamilton's
implicit function theorem, we work with an exponential map which is
not equivariant, and so the analogous computation is more
complicated, see subsection \ref{subsection:affine}. The exact
criterion for local rigidity in terms of cohomology of $\G$ with
coefficients in $\Vect^{\infty}(M)$ that results from this
computation and the implicit function theorem is Theorem
\ref{theorem:affine} which should have further applications, see
remarks at the end of subsection \ref{subsection:affine}. The main
technical difficulty in proving Theorem \ref{theorem:implicit} from
Theorem \ref{theorem:affine} is producing a tame splittings of a
sequence of tame linear maps and this is the only point at which I
use the fact that $\pi(\G)$ preserves a metric. Tameness is an
assumption on Fr\'{e}chet spaces, manifolds and maps between them
that is necessary for Hamilton's Nash-Moser implicit function
theorem \cite{Ha,Ha2}, see subsection \ref{subsection:tameness} for
discussion.

{\em Acknowledgements:} This work has benefited from conversations
with many people concerning the $KAM$ method and the related
implicit function theorems.  Of particular importance were
conversations with Jerome Benveniste, Dmitry Dolgopyat, Bassam
Fayad, Rafael Krikorian, Gregory Margulis, Peter Sarnak and
Laurent Stolovitch.  Also, thanks to Wee Teck Gan, Hee Oh, and
Gregory Margulis for conversations concerning Clozel's work on
property $\tau$, to Emmanuel Breuillard and Dmitry Dolgopyat for
discussions concerning equidistribution and tame estimates, and to
Steve Hurder and Bob Zimmer for useful comments. Special thanks to
Laurent Stolovitch for many useful comments on earlier versions on
this paper and even more useful conversations.

An early draft of this paper contained a gap. That gap is fixed by
the use, in the proof of Corollary \ref{corollary:diophantine}, of
results concerning linear forms on logarithms.  Thanks to Jordan
Ellenberg, Yuri Nesterenko, Rob Tijdeman and Michel Waldschmidt
for their help finding my way in the literature on this topic.

\section{Theorem \ref{theorem:implicit}: variants and proof}
\label{section:proofsketch}

\subsection{Background on tameness and Hamilton's implicit
function theorem}\label{subsection:tameness}

In this section we introduce the terminology of graded tame
Fr\'{e}chet spaces, tame maps and tame manifolds.  The next
several definitions are excerpted from \cite{Ha}, where they are
discussed in great detail with many examples.  The reader
interested in understanding the tame Fr\'{e}chet category cannot
do better than to study that paper.

\begin{defn}\cite[Definition II.1.1.1]{Ha}
\label{definition:graded}  A Fr\'{e}chet space is graded if the
topology is defined by sequence of semi-norms $\|{\cdot}\|_n$
where $n{\in}{\Za}^+$ such that
$\|{\cdot}\|_m{\leq}\|{\cdot}\|_{m+k}$ for any positive integers
$m$ and $k$.
\end{defn}

\noindent The reader should have in mind the seminorms on smooth
functions or vector fields defined either by the $C^n$ norms or
the $W^{2,n}$ Sobolev norms.  With either of these collections of
norms, vector fields or functions become a graded Fr\'{e}chet
space. See \cite[Section 4]{FM2} or \cite[Section 9]{Palais} for
explicit, intrinsic constructions of these norms on a Riemannian
manifold $(M,g)$. In particular, it follows from the discussion in
\cite[Section 4]{FM2} that isometries of the metric on $M$ induce
isometries of all the associated norms.  In order to be able to
use Hamilton's work, we need to use a more refined structure on
these spaces.

\begin{defn}\cite[Definitions II.1.2.1 and II.1.3.1]{Ha}
\label{definition:tamelinear} \begin{enumerate} \item A map $L$
from a graded Fr\'{e}chet space $V$ to a graded Fr\'{e}chet space
$W$ is called {\em tame} with {\em tame estimate} of degree $r$
and base $b$ if
$$\|Lv\|_k{\leq}C_k\|v\|_{k+r}$$
for all $k{\geq}b$.  Here $C_k$ is a constant depending only on
$k$ and $L$.    \item We say a graded Fr\'{e}chet space $V$ is a
{\em tame direct summand} of a graded Fr\'{e}chet space $W$ if
there are tame linear maps $F:V{\rightarrow}W$ and
$G:W{\rightarrow}V$ such that $F{\circ}G$ is the identity on $V$.
\end{enumerate}
\end{defn}

\noindent  As in the second half of Definition
\ref{definition:tamelinear}, we often omit the degree and base of
a tame linear map when they are not essential to arguments or
ideas.

We now define tame Fr\'{e}chet spaces. Given a Banach space $B$,
we let $\tilde \Sigma(B)$ be the space of sequences in $B$. We
define
$$\|\{f_k\}\|_n=\sum_{k=0}^{\infty}e^{nk}\|f_k\|_B$$
for non-negative integers $n$ and restrict our attention to the
subspace $\Sigma(B)\subset\tilde \Sigma(B)$ where $\|{\cdot}\|_n$ is
finite for all $n$.  On $\Sigma(B)$, the $\|{\cdot}\|_n$ are an
increasing family of seminorms and so makes ${\Sigma(B)}$ into a
graded Fr\'{e}chet space.   Hamilton frequently refers to
$\Sigma(B)$ as the space of exponentially decreasing sequences in
$B$, though super-exponentially decreasing is more accurate.

\begin{defn}\cite[Definitions II.1.3.2]{Ha}
\label{defn:tamefrechetspace} We call a graded Fr\'{e}chet space
$V$ a {\em tame Fr\'{e}chet space} if it is a tame direct summand
of a space $\Sigma(B)$, where $B$ is a Banach space and
$\Sigma(B)$ is equipped with the family of seminorms defined
above.
\end{defn}

\noindent For any compact manifold $M$ and any vector bundle $E$
over $M$, the space of smooth sections of $E$ is a tame Fr\'{e}chet
spaces by \cite[Theorem II.1.3.6 and Corollary II.1.3.9]{Ha}.
Hamilton's proof of these results proceeds by embedding $M$ in some
Euclidean space and using the Fourier transform.  We now sketch an
alternate, intrinsic proof of this fact.  We let $H=L^2(M,\nu,E)$ be
the space of $L^2$ sections of $E$ with $\nu$ a Riemannian volume on
$M$ defined by a fixed choice of Riemannian metric $g$, and then
form the space $\Sigma(H)$. Let $\Delta$ be the Laplacian on $E$ and
let $E_i$ be the eigenspace corresponding to eigenvalue $\lambda_i$
and let
$$V_k=\sum_{\exp(k)<1+\lambda_i}^{1+\lambda_i<\exp(k+1)}E_i.$$  If
$\sigma$ is a smooth section of $E$, then we can write
$\sigma=\{\sigma_i\}$ where each $\sigma_i{\in}E_i$ and
$$\sum_{i=0}^{\infty}(1+\lambda_i)^l\|\sigma_i\|_2<\infty$$
for all non-negative integers $l$.  In fact, this sum, for a given
value of $l$, is exactly $\|\sigma\|_{2,l}$, the $l$th Sobolev
norm of $\sigma$.  If we define $\sigma_k$ to be the component of
$\sigma$ in $V_k$, we can then define
$F:C^{\infty}(M,E){\rightarrow}\Sigma(H)$ by
$\sigma{\rightarrow}\{\sigma_k\}$ and
$G:\Sigma(H){\rightarrow}C^{\infty}(M,E)$ by taking
$\{\sigma_k\}{\rightarrow}\sum_{k=0}^{\infty}\sigma_k$.  It is
clear that $F{\circ}G$ is the identity on $C^{\infty}(M,E)$.
Tameness of both maps is immediate if we consider the Sobolev
norms on $C^{\infty}(M,E)$ and follows from the Sobolev embedding
theorems for the uniform norms.

To move out of the linear category, we need a definition of
tameness for non-linear maps.

\begin{defn}{\cite[II.2.1.1]{Ha}}
\label{definition:tamenonlinear} Let $V$ and $W$ be graded
Fr\'{e}chet spaces, $U$ an open subset of $V$ and
$P:U{\rightarrow}W$ a non-linear map.  Then we say $P$ satisfies a
{\em tame estimate} of degree $r$ and base $b$ if
$\|P(v)\|_n{\leq}C_n(1+\|v\|_{n+r})$ for all $n{\geq}b$.  Here
$C_n$ is a constant depending only on $P$ and $n$.  We say a map
$P$ is tame if it is continuous and satisfies some tame estimate
on a neighborhood of every point.
\end{defn}


The notion of a smooth tame manifold is now quite natural.  It is
a manifold with charts in a tame Fr\'{e}chet space where the
transition functions are smooth tame maps.  Similarly, a smooth
tame Lie group is a group which is a smooth tame manifold where
the multiplication and inverse maps are smooth and tame. For $M$ a
compact manifold, $\Diff^{\infty}(M)$ is proven to be a smooth
tame Lie group in \cite[Theorem II.2.3.5]{Ha}. We remark that it
is possible to see the tame structure on $\Diff^{\infty}(M)$ by
choosing an affine connection on $M$ and defining charts at the
identity as maps from $\Vect^{\infty}(M)$ defined by the
exponential map associated to the connection. This is equivalent
to what is done in \cite{Ha}.

We now recall the statement of Hamilton's implicit function
theorem for short exact sequences.

\begin{theorem}\cite[Theorem III.3.1.1]{Ha}
\label{theorem:hamilton} Let $A,B$ and $C$ be tame Fr\'{e}chet
spaces and $U,V$ and $W$ open sets in $A,B$ and $C$ respectively.
Let
$$\xymatrix{
U\ar[r]^{F}&{V}\ar[r]^{G}&{W}&\\
}$$ \noindent where $F$ and $G$ are smooth tame maps and
$G{\circ}F=0$. Suppose for each $f{\in}{U}$ the image of $DF_f$ is
the entire null space of $DG_{F(f)}$.  Further suppose there are
smooth tame maps
$$VF:U{\times}B{\rightarrow}A \hskip .5in
VG:U{\times}C{\rightarrow}B$$ \noindent such that
$VF_f:B{\rightarrow}A$ and $VG_f:C{\rightarrow}B$ are linear for
every $f{\in}U$ and
$$DF_f{\circ}VF_f+VG_f{\circ}DG_{P(f)}=\Id_B$$
for all $f{\in}U$.  Then for any $f_0{\in}U$ the image of a
neighborhood of $f_0$ fills out a neighborhood of a $g_0=F(f_0)$
in the subset of $B$ where $G(g)=0$.  Moreover, we can find a
smooth tame map $S:V'{\rightarrow}U'$ from a neighborhood $V'$ of
$g_0$ to a neighborhood $U'$ of $f_0$ such that $F{\circ}S(g)=g$
for any $g$ in $V'$ where $G(g)=0$.
\end{theorem}

\noindent The proof of this theorem is section $2$ of \cite{Ha2}.
There is a more intrinsic statement of this result as
\cite[Theorem III.3.1.2]{Ha} which does not depend on a choice of
basepoint $0$ in $C$ to write the equation $F{\circ}G=0$. In our
context the group structure on the spaces we consider defines
natural choices of basepoints and so we can work with the simpler
statement given here. In our applications, we will be able to use
a group action to reduce the problem of constructing $VF$ and $VG$
to constructing $VF_{f_0}$ and $VG_{f_0}$.  More precisely, we use
the following lemma which is essentially \cite[Lemma 4.3]{Be}, see
also \cite{Ha3}.

\begin{lemma}
\label{lemma:equivariance} Let $A,B,C,U,V,W$ and $F,G$ be as in
Theorem \ref{theorem:hamilton}.  Assume there is a smooth tame Lie
group $D$ and local smooth tame $D$ actions on $U,V$ and $W$ such
that the action on $U$ is simply transitive and $F$ and $G$ are
equivariant.  Then to obtain the conclusions of Theorem
\ref{theorem:hamilton} it suffices to produce smooth tame linear
maps $VF_{f_0}:B{\rightarrow}A$ and $VG_{f_0}:C{\rightarrow}B$
such that
$$DF_{f_o}{\circ}VF_{f_0}+VG_{f_0}{\circ}DG_{F(f_0)}=\Id_B.$$
\end{lemma}

\noindent The point of the lemma is that one need only find the
tame splitting of the complex at a single point rather than in a
neighborhood.  To prove the lemma, one simple takes the tame
splitting at $f_0$ and produces a tame splitting in the
neighborhood by translating the splitting by the $D$ action.  We
refer the reader to \cite{Be} for a more detailed proof.

\subsection{A variant of Theorem
\ref{theorem:implicit} for arbitrary smooth actions}
\label{subsection:affine}
 Let $M$ be a compact manifold and  $\Vect^{\infty}(M)$ the graded Fr\'{e}chet
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
$\Diff^{\infty}(M){\subset}C^{\infty,\infty}(M)$. 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 our purposes it always suffices to work with the
Levi-Civita connection defined by some metric.

For the remainder of this section, we 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 $R$ 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 and their formal inverses.  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 $w_i$
as a word in $k$ letters and taking $(\psi_1,{\ldots},\psi_k)$ to
$(w_1(\psi_1,{\ldots},\psi_k),{\ldots},w_r(\psi_1,{\ldots},\psi_k))$.
Letting $\Id$ be the identity map on $M$, it follows that
$P(\Id)=\pi$ and $Q(\pi)=(\Id,{\ldots},\Id)$. Note that
$Q{\inv}(\Id,{\ldots},\Id)$ is exactly the space of $\G$ actions.
While this is a closed subset of $\Diff^{\infty}(M)^k$, it is
unclear that it is a manifold or even a union of manifolds in a
neighborhood of any point. In the finite dimensional setting, this
set is an algebraic variety, and therefore a manifold at ``most"
points.

The tangent spaces of $\Diff^{\infty}(M)$ at any point is
identified with $\Vect^{\infty}(M)$.  To avoid notational
confusion, we let $A=\Diff^{\infty}(M), B=\Diff^{\infty}(M)^k$ and
$C=\Diff^{\infty}(M)^r$. Note that $A,B$ and $C$ are tame
manifolds. 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}
By Theorem \ref{theorem:hamilton}, local rigidity follows if there
exist 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
$VQ:U{\times}\Vect^{\infty}(M)^r{\rightarrow}U{\times}\Vect^{\infty}(M)^k$
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 Lemma \ref{lemma:equivariance}
above.  I.e. we need only find a splitting of the sequence:

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

\noindent Note that to this point in the discussion we are not
assuming that $\pi(\G)$ preserves any geometric structure of any
kind.  The discussion so far yields the following technical
result.

\begin{proposition}
\label{proposition:fromhamilton} Let $\G$ be a finitely presented
group, $M$ be a compact manifold,
$\pi:\G{\rightarrow}\Diff^{\infty}(M)$ be a homomorphism,  and let
$P$ and $Q$ be the maps defined at the beginning of this section.
Then $\pi$ is locally rigid provided the tame linear complex in
$(\ref{equation:splittingsequencepoint})$ is tamely split.  In
other words, the action is locally rigid provided there exist maps
$VP:\Vect^{\infty}(M)^k{\rightarrow}\Vect^{\infty}(M)$ and
$VQ:\Vect^{\infty}(M)^r{\rightarrow}\Vect^{\infty}(M)^k$ such that
$$VP{\circ}DP_{\Id}+VQ{\circ}DQ_{\pi}=\Id_{\Vect^{\infty}(M)^k}.$$
\end{proposition}

\noindent {\bf Remark:} To apply Theorem \ref{theorem:hamilton},
we are identifying a neighborhood of the identity in
$\Diff^{\infty}(M)$ with a neighborhood of $0$ in
$\Vect^{\infty}(M)$ via the map $\Exp$. This identification
naturally identifies the point $(\Id,{\ldots},\Id)$ in $C$ with
the $0$ in $\Vect^{\infty}(M)^r$.


We now compute the derivatives $DP_{\Id}$ and $DQ_{\pi}$
explicitly. This relates the sequence in line
$(\ref{equation:splittingsequencepoint})$ to the sequence of
$\Vect^{\infty}(M)$ valued cochains on $\G$.  We identify the
cohomology of $\G$ with the simplicial cohomology of a simplicial
$K(\G,1)$ space with one vertex, one edge for each generator of
$\G$ and one two cell for each relator of $\G$.   We will write
$C^i(\G,\Vect^{\infty}(M))$ for $i$-cochains on $\G$ with values
in $\Vect^{\infty}(M)$.  One can identify $0$-cochains
$C^0(\G,\Vect^{\infty}(M))$ with $\Vect^{\infty}(M)$,
$1$-cochains, $C^1(\G,\Vect^{\infty}(M))$, with maps from $S$ to
$\Vect^{\infty}(M)$ or equivalently $\Vect^{\infty}(M)^k$, and
$2$-cochains, $C^2(\G,\Vect^{\infty}(M))$, with maps from $R$ to
$\Vect^{\infty}(M)$ or equivalently $\Vect^{\infty}(M)^r$.   In
these identifications, the (co)differentials $d_0$ and $d_1$ can
be written explicitly as
$$d_0(v)=(v-d\pi(\g_j)v)_{j{\in}J}$$ and
$$d_1(f)=\left(\sum_{k_i=1}^{l(w_i)}\left(\prod_0^{k_i-1}d\pi(\g_{k_i})\right)f(\g_{k_i})\right)_{i{\in}I}$$
where $w_i=\prod_{k_i=1}^{l(w_i)}\g_{k_i}$ and we interpret the
empty product as $d\pi(e)$ where $e{\in}\G$ is the identity.  We now
want to show that the complex in
$(\ref{equation:splittingsequencepoint})$ is identical to the
complex:
$${\xymatrix{
{C^0(\G,\Vect^{\infty}(M))}\ar[r]^{d_0}&{C^1(\G,\Vect^{\infty}(M))}\ar[r]^{d_1}&{C^2(\G,\Vect^{\infty}(M))}&\\
}}$$ \noindent Since we already have identifications of spaces
$C^0(\G,\Vect^{\infty}(M))\simeq\Vect^{\infty}(M)$,
$C^1(\G,\Vect^{\infty}(M))\simeq\Vect^{\infty}(M)^k$ and
$C^2(\G,\Vect^{\infty}(M))\simeq\Vect^{\infty}(M)^r$, it suffices
to prove the following:

\begin{proposition}
\label{proposition:cohomology} With the identifications above, we
have
\begin{equation}
\label{equation:d1} DP_{\Id}(v)=d_0(v) \end{equation}
 and
\begin{equation}
\label{equation:d2} DQ_{\pi}(v_1,\cdots,v_k)=d_1(f)
\end{equation}
\noindent where $v_i=f(\g_i)$ if
$f{\in}C^1(\G,\Vect^{\infty}(M))$.
\end{proposition}

\noindent To prove the proposition, we need some elementary facts
concerning the exponential map.  We fix an affine connection
$\nabla$ on $M$, we choose a cover $\mathcal U$ of $M$ by normal,
convex neighborhoods, i.e. neighborhoods such that any two points
in the neighborhood are joined by a geodesic lying entirely in the
neighborhood.  There exists an $\varepsilon>0$ such that we can
choose a further cover $\mathcal V$ such that for any
$V{\in}\mathcal V$  there is a neighborhood $U{\in}\mathcal U$
such that for any $X\in\Vect^{\infty}(M)$ with
$\|X\|_{C^1}<\varepsilon$ the maps $x{\rightarrow}x+X_x$ and
$x{\rightarrow}\exp_xX_x$ define maps from $V$ to $U$. In what
follows, we use this fact to write the difference of two nearby
diffeomorphisms as a vector field.

\begin{lemma}
\label{lemma:quadraticerror} Let $X,Y{\in}\Vect^{\infty}(M)$ and
$V{\subset}U$ as described above.  Then for $t$ small enough,
$\Exp(tX)\circ\Exp(tY)=\Exp(t(X+Y))+t^2Z$ on $V$ where $Z$ is some
vector field on $U$ with $\|Z\|_n{\leq}C\max(\|X\|_{n+1},\|Y\|_n)$.
\end{lemma}

\noindent {\bf Proof.} This is a consequence of the defining
equation of geodesics and some elementary calculus.  Let
$(x_1,\ldots,x_n)$ be coordinates on a neighborhood $U$ in $M$ and
$(x_1,\ldots,x_n,p_1,{\ldots},p_n)$ coordinates on $TU$.  Recall
that the geodesic equation can be written as the pair of coupled
equations:

$$\frac{dx_k}{dt}=p_k$$

\noindent and

$$\frac{dp_k}{dt}=-\sum_{i,j}\Gamma^k_{ij}p_ip_j$$

\noindent where $\Gamma^k_{ij}$ are the Christoffel symbols for the
connection in the local coordinate system. From these formulas, it
is clear that up to a factor that is quadratic in $t$ geodesics are
straight lines in the coordinates. The change in $X$ from $m$ to
$\Exp(Y)(m)$ is of order $t$ which yields another quadratic error
and also accounts for the need to control $\|Z\|_n$ by
$\|X\|_{n+1|}$. \qed

\noindent{\bf Remark:} Lemma \ref{lemma:quadraticerror} implies
that $\Exp(tX)\Exp(-tX)$ is quadratic in $t$ and so
$\Exp(tX){\inv}$ is equal to $\Exp(-tX)$ to first order in $t$.

\begin{lemma}
\label{lemma:differencequotient} For any
$X{\in}\Vect^{\infty}(M)$,
$$\lim_{t{\rightarrow}0}\frac{\Exp(tX)}{t}=X.$$
\end{lemma}

\noindent {\bf Proof.} Again this is a straightforward computation
from the geodesic equation which implies that the difference
between $\Exp(tX)x$ and $x+(tX)_x$ is quadratic in $t$. \qed

\begin{lemma}
\label{lemma:notaffine} For any $X{\in}\Vect^{\infty}(M)$ and any
$\phi{\in}\Diff^{\infty}(M)$, we have that
$\Exp(td\phi(X))=\phi{\circ}\Exp(tX)\circ{\phi}{\inv}+t^2Z$ where
$Z$ is some vector field with $\|Z\|_n{\leq}C\|X\|_n$ where $C$
depends on $M$ and $\phi$.
\end{lemma}

{\noindent}{\bf Remarks:}\begin{enumerate} \item If $\phi$
preserves the connection defining $\Exp$, then we can in fact
choose $Z=0$, i.e. $\Exp$ is equivariant. \item The lemma is not
stated entirely precisely, one should make choices of $U$ and $V$
as in Lemma \ref{lemma:quadraticerror}, and these choices will
also depend on $\phi$.
\end{enumerate}

\noindent {\bf Proof.} This fact can be deduced from standard
computations of the derivative of left and right multiplication in
$\Diff^{\infty}(M)$, see e.g. \cite[Example I.4.4.5]{Ha}. The
point is that the right hand side is $\Exp'(tX)$ where $\Exp'$ is
the exponential map defined by pushing forward the connection via
$\phi$.  It is easy to see from the geodesic equations as in the
proof of \ref{lemma:quadraticerror} that the difference between
$\Exp'(tX)$ and $\Exp(tX)$ is quadratic in $t$. \qed

\noindent {\bf Proof of Proposition \ref{proposition:cohomology}.}
By definition
$$DP_{\Id}(X)=\lim_{t{\rightarrow}0}\frac{P(tX)-P(\Id)}{t}.$$
\noindent Note that
$P:\Diff^{\infty}(M){\rightarrow}\Diff^{\infty}(M)^k$ is defined
component by component, i.e. $P=(P_1,{\ldots}P_k)$ where
$P_i:\Diff^{\infty}(M){\rightarrow}\Diff^{\infty}(M)$ is given by
$P_i(\phi)=\phi\pi(\gamma_i)\phi{\inv}$.  It therefore suffices to
compute $DP_i$.  We have
$$DP_i({\Id})(X)=\lim_{t{\rightarrow}0}\frac{P_i(tX)-P_i(\Id)}{t}$$
$$=\lim_{t{\rightarrow}0}\frac{\Exp(tX)\pi(\gamma_i)\Exp(tX){\inv}-\pi(\gamma_i)}{t}.$$
Since we identify $T(\Diff^{\infty}(M))_{\pi(\gamma)}$ with
$\Vect^{\infty}(M)$ by multiplying on the right by
$\pi(\gamma){\inv}$ this is equivalent to
$$DP_i({\Id})(X)=\lim_{t{\rightarrow}0}\frac{\Exp(tX)\pi(\gamma_i)\Exp(tX){\inv}\pi(\gamma_i){\inv}}{t}$$
$$=\lim_{t{\rightarrow}0}\frac{\Exp(tX)\Exp(td\pi(\gamma_i)X){\inv}}{t}$$
$$=\lim_{t{\rightarrow}0}\frac{\Exp(tX-td\pi(\gamma_i)X)}{t}$$
by where the second equality follows from Lemma
\ref{lemma:notaffine} and the third from Lemma
\ref{lemma:quadraticerror}.  The limit is equal to
$X-d\pi(\gamma_i)X$ by Lemma \ref{lemma:differencequotient}. This
proves equation $(\ref{equation:d1})$.

The manipulation to prove equation $(\ref{equation:d2})$ is
similar so we only sketch it. As before, we can write
$Q=(Q_1,{\ldots},Q_r)$ where
$Q_i(\phi_1,\ldots,\phi_k)=w_i(\phi_1,\ldots,\phi_k)$.  This
yields that
$$DQ_i({\pi})(X_1,\ldots,X_k)=$$
$$\lim_{t{\rightarrow}0}\frac{Q_i(\Exp(tX_1)\pi(\gamma_1),\ldots
,\Exp(tX_k)\pi(\gamma_k))-Q_i(\pi(\gamma_1),\ldots,\pi(\gamma_k))}{t}$$
$$=\lim_{t{\rightarrow}0}\frac{w_i(\Exp(tX_1)\pi(\gamma_1),\ldots
,\Exp(tX_k)\pi(\gamma_k))-w_i(\pi(\gamma_1),\ldots,\pi(\gamma_k))}{t}$$
$$=\lim_{t{\rightarrow}0}\frac{w_i\big(\Exp(tX_1)\pi(\gamma_1),\ldots
,\Exp(tX_k)\pi(\gamma_k)\big)w_i\big(\pi(\gamma_1),\ldots
,\pi(\gamma_k)\big){\inv}}{t}.$$ All further manipulations to
complete the proof use manipulations similar to those above using
Lemmas \ref{lemma:quadraticerror}, \ref{lemma:differencequotient}
and \ref{lemma:notaffine}.  The proof can also be given in the
formalism of the so-called Fox calculus. \qed

\noindent Combining Proposition \ref{proposition:cohomology} and
Proposition \ref{proposition:fromhamilton}, we have 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
$${\xymatrix{
{C^0(\G,\Vect^{\infty}(M))}\ar[r]^{d_0}&{C^1(\G,\Vect^{\infty}(M))}\ar[r]^{d_1}&{C^2(\G,\Vect^{\infty}(M))}&\\
}}$$ \noindent admits a tame splitting, then the homomorphism
$\pi$ is locally rigid.
\end{theorem}

\smallskip \noindent {\bf Remarks on further applications:}
In the context of Theorem \ref{theorem:affine}, the general
question of producing splittings seems difficult. However, if $\G$
is a cocompact lattice in a semisimple Lie group with no compact
factors and $\G$ has property $(T)$ of Kazhdan, the action of $\G$
on $M$ is affine and $M$ is parallelizable, then it seems likely
that one will be able to produce a tame splitting of the sequence
in Theorem \ref{theorem:affine} using estimates on Laplacians
produce by variants and generalizations of the Bochner method as
in \cite{FH}. This yields results similar to those in \cite{FM3}
for actions of cocompact lattices in $SP(1,n)$ and $F_4^{-20}$ as
well as the (cocompact) higher rank lattices considered in
\cite{FM3}. This is work in progress, joint with T.Hitchman. To
prove theorems as in \cite{FM3} for non-cocompact lattices by
these methods seems quite difficult, except perhaps when the
lattices have $\Qa$ rank $1$.

\subsection{Reductions for the proof of Theorem \ref{theorem:implicit}}
\label{subsection:theorem1.1}

In this section, we reduce the proof Theorem
\ref{theorem:implicit} from Theorem \ref{theorem:affine} to
certain estimates that we prove in section
\ref{section:estimates}. For this purpose, we now assume $g$ is a
Reimannian metric on $M$ and that
$\pi:\G{\rightarrow}\Isom(M,g){\subset}\Diff^{\infty}(M)$.

Since we are assuming that $H^1(\G,\Vect^{\infty}(M))=0$, so that
the map $d_0:{C^0(\G,\Vect^{\infty}(M))}{\rightarrow}{\Ker(d_1)}$ is
surjective.  To define a splitting we write $\Vect^{\infty}(M)$ as a
Hilbertian direct sum of $\pi(\G)$ invariant finite dimensional
subspaces $V_j$.  To do this, we apply the Peter-Weyl theorem to the
closure $K$ of $\pi(\G)$ in $\Isom(M,g)$ which is compact. The
representation of $K$ on $\Vect_{2}(M)$ or $\Vect_{2,k} (M)$ is
therefore a Hilbertian sum of irreducible $K$ (and therefore $\G$)
modules. Since the action of $\Isom(M,g)$ commutes with the
Riemannian Laplacian $\Delta$ on $M$, each $V_j$ is contained in an
eigenspace for the Laplacian, with eigenvalue $\lambda_j$. By
standard elliptic theory, this implies that each $V_j$ consists of
smooth functions and that the splitting into $V_j$'s does not depend
on $k$. We split the complex in Theorem \ref{theorem:affine} by
splitting each complex in the sequence of complexes:

\begin{equation} \label{equation:splittingfdsequence}
 {\xymatrix{
{V_j}\ar[r]^{d_0}&{V_j^k}\ar[r]^{d_1}&{V_j^r}&\\
}}
\end{equation}

\noindent For each $V_j$ we let $d_i^j$ be the restriction of $d_i$
to $V_j$. We define $(D_1^j)$ to be orthogonal projection onto
$\im(d_0^j)=\ker(d^j_1)$ followed by $(d_0^j){\inv}$, the inverse of
$d_0^j$ viewed as an isomorphism from $\ker(d_0^j)^{\perp}$ to
$\im(d_0^j)=\ker(d^j_2)$. Similarly, we define $(D_2^j)$ to be
orthogonal projection onto $\im(d_1^j)$ followed by $(d_1^j){\inv}$,
the inverse of $d_1^j$ viewed as an isomorphism from
$\ker(d_1^j)^{\perp}$ to $\im(d_1^j)$. This gives a splitting of
equation (\ref{equation:splittingfdsequence}). To split the sequence
in Theorem \ref{theorem:affine}, we let $D_i(\sum_j v_j)= \sum_j
(D_i^j)(v_j)$ . At the moment this map is only a formal splitting,
since it is not clear that if $\sum_J v_j$ is smooth that
$D_i(\sum_j v_j)$ is smooth or even in $L^2$. To verify that the
maps $D_i$ maps smooth cochains to smooth cochains and that the
splitting is tame, we require upper bounds on $(D_i^j)$. Or
equivalently, lower bounds on $d_i^j$. The maps $D_i$ we have
defined are well-defined without estimates on elements that project
non-trivially into only finitely many eigenspaces.

We will work with the Sobolev norms $\|{\cdot}\|_{2,k}$ on
$\Vect^{\infty}(M)$. Since the $C^k$ norms are tamely equivalent to
the Sobolev norms this is equivalent to considering the $C^k$ norms.
A key point is that 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 $\Vect^{\infty}(M)$, 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 an
equivalent definition for the Sobolev norm, and the second equality
is the fact that $v{\in}V_j$.

We now give a sufficient criterion for tame splitting.

\begin{proposition}
\label{proposition:lowerbounds} For $D_1$ and $D_2$ to define a
tame splitting of the sequence in Theorem \ref{theorem:affine}, it
suffices to find $\epsilon>0$ and $\alpha$ a positive integer such
that
$$\|d_i^j(v)\|\geq \frac{\epsilon}{\lambda_j^{\alpha}}\|v\|$$
\noindent on $\ker(d_i^j)^{\perp}$ for $i=1,2$.
\end{proposition}

\noindent{\bf Proof} The fact that $d_0 \circ D_1 + D_2 {\circ} d_1
=\Id$ formally is immediate from the definitions, so all that
remains is to check that $D_1$ and $D_2$ map smooth chains to smooth
chains and are tame.  If $v$ is a smooth $i$-chain and $v=\sum v_j$
with $v_j{\in}V_j$ then, up to absolute constants, the Sobolev norm
$\|v\|_{2,n}$ is $\sum \lambda_j^n\|v_j\|_2$ for every $n$.  We can
view $D_i$ as a composition of a projection $\pi_i$ and the map
$d_{i-1}{\inv}=\sum (d_{i-1}^j){\inv}$ which is defined on the image
of the projection. The projections are tame by definition, so we
restrict to chains in the image of the projections.  For such a
chain $v=\sum_j v_j$ we have $d_i^j((d_i^j){\inv}(v_j))=v_j$ and
substituting $w_j=(d_i^j){\inv}(v_j)$ into the hypotheses on
$d_i^j$, we see our hypothesis is equivalent to:

$$\|(d_i^j){\inv}(v)\|_2{\leq}\frac{\lambda^{\alpha}}{\epsilon}\|v\|_2.$$

\noindent Combined with our earlier observation on Sobolev norms,
this implies that

$$\|d_i{\inv}(v)\|_{2,n} \leq C\sum_j \lambda^n \|(d_i^j){\inv}(v)\|_2
\leq \frac{C'}{\epsilon}\sum_j \lambda^{n+\alpha} \|v\| \leq C''
\|v\|_{2, n + \alpha}$$

\noindent where $C''$ depends only on $\epsilon$. \qed


\smallskip \noindent {\bf Remarks on finite regularity and relation to KAM method:} One can write down a sequence like that in line
$(\ref{equation:initialsequence})$ with $\Diff^l(M)$ in place of
$\Diff^{\infty}(M)$, but in that context the maps $P$ and $Q$ will
not be smooth, and so no implicit function theorem will apply.
Lack of smoothness will derive from the fact that for any smooth
structure on $\Diff^l(M)$, only one of right and left
multiplication is smooth, see \cite[Examples I.4.4.5 and
I.4.4.6]{Ha}. This makes this approach seem unlikely to yield
results in finite regularity, in particular the finite regularity
version of Theorem \ref{theorem:isomrigid} from \cite{FM2} seem
hard to prove by this method.

It should be possible to prove a finite regularity version of
Theorem \ref{theorem:implicit} by replacing the use of the
implicit function theorem with an explicit $KAM$-type iteration.
It is well known that the KAM iteration method and the implicit
function theorem are intimately connected.  Even if successful,
this method seems unlikely to yield as sharp a finite regularity
version of Theorem \ref{theorem:isomrigid} as obtained in
\cite{FM2}. For related applications of the $KAM$ method, see
\cite{DK,DoKr}.

It is also possible to work strictly in finite regularity and
prove cohomological criteria for local rigidity, see \cite{AN,Fl}.
The difficulty with these results is that the resulting
cohomological conditions are more difficult to verify and in
particular do not yield a finite regularity analogue of Theorem
\ref{theorem:implicit}.

\section{Tame estimates for the proof of Theorem
\ref{theorem:implicit}} \label{section:estimates}

In this section we complete the proof of Theorem
\ref{theorem:implicit} by veryifying the hypotheses of Proposition
\ref{proposition:lowerbounds}. In the first subsection, we see that
$H^1(\Gamma,\Vect^{\infty}(M))=0$ has strong implications for the
group $\pi(\Gamma)$. These constraints are used in the second and
third subsections respectively to prove the desired bounds on $d_0$
and $d_1$ respectively.

\subsection{Structure of actions with $H^1(\Gamma,
\Vect^{\infty}(M))=0$}

Given $\pi:\Gamma{\rightarrow}\Isom(M)$, we let $K =
\overline{\pi(\Gamma)}$.  The group $K$ is a compact group with
connected component $K^0$, so $K/K^0$ is finite. There is a largest
finite index normal subgroup $\Gamma^0<\Gamma$ with
$\pi(\Gamma^0)<K^0$. We denote by $\mathfrak k$ the lie algebra of
$K^0$ and note $\mathfrak k \subset \Vect^{\infty}(M)$ simply by
differentiating the $K^0$ action. It is also clear that $\mathfrak
k$ is a $\pi(\Gamma)$ invariant subspace of $\Vect^{\infty}(M)$. The
$\pi(\Gamma)$ action on $\mathfrak k$ is the restriction of the
adjoint action.  The following proposition follows from the fact
that $H^1(\Gamma, \mathfrak k)=0$ and known results concerning group
homomorphisms and compact groups. We use the convention that a
connected compact group is called {\em semisimple} if it has no
circle factors.

\begin{proposition}
\label{proposition:structure} Under the hypotheses of Theorem
\ref{theorem:implicit}, $K_0$ is semisimple. Furthermore, there is a
$\Qa$ structure on $K_0$ such that:
\begin{enumerate}
\item all representations of $K_0$ are defined over $\Qa$ and,
\item possibly after conjugating the action by an element of
$K_0$, there is a finite algebraic extension $k$ of $\Qa$ such
that all elements of $\pi(\Gamma){\cap}K_0$ have entries in $k$.
\end{enumerate}
\end{proposition}

\noindent{\bf Proof:} We first show semisimplicity of $K^0$
assuming $K=K^0$. We assume $K^0$ is not semisimple and derive a
contradiction. If $K^0$ is not semisimple, then $K^0=K^0_sT$ where
$K^0_s$ is semisimple, $T$ is a torus, $T{\cap}K^0_s=D$ a finite
abelian group, and the product is an almost direct product. The
$K^0_s$ invariant vector fields on $M$ clearly include a copy of
$\mathfrak t$ on which the $\Gamma$ action is trivial. This set is
non-empty since it contains the Lie algebra $\mathfrak t$. The
homomorphism $\pi:\Gamma{\rightarrow}K^0_S$ clearly yields a
homomorphism $\pi:\Gamma{\rightarrow}T/D$ with dense image. This
homomorphism clearly admits deformations which implies that
$H^1(\Gamma,\mathfrak t)\neq0$, a contradiction.

Let $F=K/K^0$.  There is a natural $F$ action on $K^0$ orbits in
$M$ and denote the kernel of this action by $F'$. Elements of $K$
that project to $F'$ map each $K^0$ orbit in $M$ to itself.
Assuming $F=F'$, we can repeat almost the same argument as above.
Here $F'$ acts on the torus $T$ by automorphisms and hence by
permutations of the coordinates.  Taking an appropriate diagonal
subgroup of $\Delta<T$, we have a trivial $F'$ action.  We can now
project $\pi(\Gamma)$ to $\Delta/{\Delta{\cap}D}$ and argue as
above.

In the general case $F' \neq F$, we reduce back to the previous
setting. Let $\Gamma'$ be the inverse image of $K'$ in $\Gamma$. We
recall that there is a set U whose complements consists of closed
submanifolds of codimension at least 1 which is diffeomorphic to
$K^0/C{\times}S$ on which the $K^0$ action is by left translations.
In particular, we can choose a open, $K'$ invariant subset $U'$
containing at least $\frac{1}{2}$ the measure of $M$. Arguing as in
the last paragraph, on the subset of $V=\Vect_2(M)$ supported on
$U'$, we can find a non-zero class represented by smooth vector
fields in $H^1(\Gamma',V)$.   The $\Gamma$ (resp. $K$)
representation on $\Vect(M)$ contains the representation $\pi_I$
induced from the $\Gamma'$ (resp. $K'$) representation on $V$. Since
we are dealing with compact groups, we can write the representation
on $\Vect_2(M)$ as direct sum of $\pi_I$ and a complementary
representation.  This implies that the cocycle defined above gives
rise to a non-vanishing cocycle in $H^1(\Gamma,\Vect_2(M))$.  It is
straightforward from the definition of induction that the cocycle is
represented by smooth vector fields.

That there is a $\Qa$ form of $K_0$ for which all representations
are defined over $\Qa$ is in \cite[Section 3]{Rg2} and a more
general result can be found as \cite[Theorem 1.2]{M1}.  It follows
easily that we can represent $K$ as a $\Qa$ group.

By hypothesis $H^1(\Gamma,\mathfrak k)=0$ and so $\Gamma$ is
locally rigid in $K$ by the main result of \cite{We3}. The fact
that after conjugating by an element of $K^0$, the elements of
$\pi(\Gamma)$ have entries in a number field $k$ then follows from
\cite[Proposition 6.6]{Rg}. We remark that to see that the entries
of $\pi(\Gamma^0)$ are defined over a number field $k$, one can
avoid checking $K$ is defined over $\Qa$ by noting that the proof
of \cite[Proposition 6.6]{Rg} only uses the algebraic structure on
the connected component of $K$. This suffices for our
purposes.\qed


\begin{corollary}
\label{corollary:diophantine} Under the hypotheses of Theorem
\ref{theorem:implicit} for any $r>0$ there exists $\alpha \geq 0$
and $\epsilon
>0$ such that for any $\gamma{\in}B_{\Gamma}(e, r)$ we have

$$\|v-d\pi(\gamma)v\|\geq
\frac{\epsilon}{\lambda^{\alpha}}\|v\|$$


\noindent for any $v{\in}V_{\lambda}$ orthogonal to the set where
$w=\pi(\gamma)w$.
\end{corollary}

\noindent{\bf Proof:} We begin by assuming that $\pi(\gamma)$ is in
connected component of $\overline{\pi(\Gamma)}$ and end by reducing
to this case.

Since $\gamma{\in}K_0$, a connected semisimple compact group,
$\gamma$ is contained in some maximal torus $T<K_0$.  We recall
that Proposition \ref{proposition:structure} implies that all
$K_0$ representations are rational and that $\gamma$ has entries
in a number field $k$. The action of $\gamma$ on $\Vect_2(M)$ is
the restriction of the $K_0$ action and/or the $T$ action. We
briefly recall a few facts from the representation theory of
compact groups. First, since $T$ is abelian, the irreducible
representations of $T$ are the usual complex dimension one
representations which can identified with $\Za^d$.  The subset of
these representations that occur in a $K_0$ representation are
exactly those that are invariant under the induced action of Weyl
group of $K_0$ on $\Za^d$, see e.g \cite[Proposition VI.2.1]{BD}.
This subset is a sublattice $\Za^k$ in $\Za^d$. We can decompose
the action of $T$ in any representation of $K_0$ into $2$
dimensional subspaces where ${\bf t}{\cdot}v_{\bf l}=exp(\iota
2\pi<{\bf t},{\bf l}>)v_{\bf l}$ where ${\bf l}{\in}\Za^k$ and the
inner product is defined by taking any element of $\Ra^d$ covering
$t$ in $\Ta^d$ in the usual covering map.  In particular, the same
formula holds for ${\bf t}={\bf \pi(\gamma)}$.

To determine the behavior of $\pi(\gamma)$ we choose a basis ${\bf
e_1, \cdots, e_k}$ for $\Za^k$. Note that the basis vectors occur
in some finite collection of representations of $K_0$ all of which
are rational by Proposition \ref{proposition:structure}. Since
$\pi(\gamma)$ can be chosen to lie in some number field $k$, the
dimension of the $K_0$ representations containing $v_{e_i}$ is
bounded and $exp(\iota 2\pi\langle{\bf t},{\bf e_i}\rangle)$ is an
eigenvalue in that representation, it follows that each
$\alpha_i=exp(\iota 2\pi\langle{\bf \pi(\gamma)},{\bf
e_i}\rangle)$ lies in a number field $L$. Writing
$\theta_i=(\langle {\bf \pi(\gamma)},{\bf e_i}\rangle)$  we see
that $\theta_i= \frac{\log(\alpha_i)}{\log(-1)}$ where $\alpha_i$
and $-1$ are both algebraic.  Results on linear forms in
logarithms, see e.g. \cite[Theorem 3.1]{Bak} and discussion
following, imply that each $\theta_i$ is diophantine. We prove the
desired estimate using a variant of the same theorem. We have that

$$\|d\pi(\gamma)v_{\bf l}-v_{\bf l}\|=|exp(\iota
2\pi\langle{\bf \pi(\gamma)},{\bf l}\rangle)-1|\|v_{\bf l}\|$$

\noindent so it suffices to bound $|exp(\iota 2\pi\langle{\bf
\pi(\gamma)},{\bf l}\rangle)-1|$ below by $\frac{C}{\|k\|^d}$.
This is equivalent to bounding the term $(\iota 2\pi\langle{\bf
\pi(\gamma)},{\bf l}\rangle)$ away from integers by
$\frac{C'}{\|k\|^d}$. This follows as
$$|p-\sum_i 2 l_i
\frac{\log(\alpha_i)}{\log(-1)}|>C|p \log(-1) - \sum_i 2 l_i
\log(\alpha_i)|>\frac{C'}{\|{\bf l}\|^d}$$

\noindent unless the left hand side is zero, in which case $v_l$
is $\pi(\gamma)$ invariant by definition.  Here we are using
\cite[Theorem 3.1]{Bak} and the discussion on the following page
to bound the linear form in logarithms below by
$\exp(-\log(h){\cdot}C)$ where $h$ is a bound on the heights of
the set $p,l_1, \ldots, l_k$. The height of an integer is just
it's absolute value and for the quantity on the left hand side to
be small, the absolute value of $p$ can be no more than a constant
times the maximal height of the $l_i$'s.

It remains to show a relationship between ${\bf l}$ and $\lambda$.
Let $w{\in}{\mathfrak k_0}$ be the vector field on $K_0$ such that
$\exp(w)=\pi(\gamma)$.  Using the $K_0$ action on $M$ and viewing
$v_{\bf l}$ as a vector field on $M$, we can differentiate $v_{\bf
l}$ by $w$ and note that $\|w{\cdot}v_{\bf l}\|= \prod_i |l_i|
\|v_{\bf l}\|$ point-wise. This then implies that the $C^1$ norm
of $v_{\bf l}$ is bounded below by $\|{\bf l}\|$ times the $C^0$
norm of $v_{\bf l}$. This and the definition of Sobolev norm imply
that:

$$\|{\bf l}\|\|v_{\bf l}\|_2\leq\|{\bf l}\|\|v_{\bf l}\|_{C^0}{\leq}
\|v_{\bf l}\|_{C^1}\leq C(1+\lambda)^\frac{dim(M)}{2}\|v_{\bf
l}\|_2.$$

\noindent Simplifying, we see that $\|{\bf l}\|\leq C \lambda ^{d'}$
where $d'=\frac{\dim(M)}{2}$ where $C$ is a constant depending only
on $M$.  Therefore we have


\begin{equation}
\label{equation:diophantine} \|d\pi(\gamma)v_{\bf l}-v_{\bf
l}\|{\geq}\frac{\epsilon}{\lambda^\alpha}
\end{equation}

\noindent for some choice of $\epsilon$ and $\alpha$ which depend
on $L,M$ and $\gamma$.

The conclusion of the theorem follows for $K=K_0$, since we
consider only finitely many elements of $\Gamma$.  To prove the
general case, we note that there is a number $n$ such that for any
$\gamma{\in}\Gamma$ we have $\pi(\gamma^n){\in}K_0$.  This implies
equation $(\ref{equation:diophantine})$ holds up to replacing
$\epsilon$ by $\frac{\epsilon}{n}$ on any subspace where $K_0$
acts non-trivially. Since $K/K^0$ is finite, a stronger bound
holds trivially on $K_0$ invariant vectors, with $\alpha=0$ and
$\epsilon$ depending only on $S$ and $F$. \qed

\subsection{Lower bounds on $d_0$}
\label{subsection:dp}

We remark that the required lower bound on $d_0$ are an immediate
consequence of Corollary \ref{corollary:diophantine}. In this
subsection, we explain how to adapt the ideas in \cite{Do} to prove
lower bounds for $d_0$ without using the results on linear forms in
logarithms referred to in the proof of Corollary
\ref{corollary:diophantine}. The bound obtained in this way is both
easier and involves a much smaller loss of regularity. Let $K$ be a
compact group whose connected component is semi-simple and $\G$ a
finitely generated dense subgroup of $K$. Assume that $K$ acts on a
compact manifold $M$. The action preserves some metric $g$ and
therefore commutes with the Laplacian $\Delta$ defined by $g$. As
above, we can write the space of $L^2$ vector fields on $M$ as a
Hilbertian direct sum of subspaces $V_j$ where each $V_j$ is an
irreducible $K$-module and is contained in an eigenspace for
$\Delta$ with eigenvalue $\lambda_j$. We then have the following,
which is a generalization and sharpening of results in
\cite[Appendix A]{Do}.

\begin{theorem}
\label{theorem:dolgopyatplus} There exist $\varepsilon_0>0$ 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
$$\|v-d\pi(\g)v\|_2{\geq}\varepsilon_0(\log(1+\lambda_j))^{-4}\|v\|_2.$$
\end{theorem}

\noindent{\bf Remarks:} \begin{enumerate} \item This says implies
that $d_0{\inv}$ is tame whenever the closure of $\G$ in
$\Isom(M,g)$ has semisimple connected component. In fact, it
implies the estimate on $d_0$ for Proposition
\ref{proposition:lowerbounds} with $\alpha=1$.

\item In \cite{Do}, this result is proven with for the action on
$L^(M)$ when $M=K/C$ with $\lambda^{-\alpha}$ in place of
$(\log(1+\lambda_j))^{-4}$. The improvement in estimate here comes
primarily from using the results from \cite[Appendix 3]{CN} in
place of those in \cite{Do}.  It is actually possible to replace
the $-4$ in the exponent above with something closer to $-2$. This
is unimportant to our applications.  See below for further
discussion.
\end{enumerate}

\noindent Before proving the theorem, we recall a fact from
\cite[Appendix 3]{CN}.  Since we have fixed a finite generating
set $S$ for $\G$, we can look at sets $B(\G,n)$ which consist of
elements of $\G$ that can be written as words of length less than
$n$ in $S$.

\begin{proposition}
\label{proposition:fromdima}  There exists a constant $C$ such
that for any $\varepsilon>0$ and any $k{\in}K$ the set of points
$B(\G,n)k$ form an $\varepsilon$-net in $K$ where
$n=\varepsilon_0(\log(\frac{1}{\varepsilon}))^4$.
\end{proposition}

\noindent{\bf Remarks:}\begin{enumerate} \item In \cite{CN}, this
result is only stated for the case $K=SU(2)$, though it is later
remarked that it holds for $SU(n)$ as well.  The fact that it
follows from the same proof for all $K$ we are considering here was
first observed by Michael Larsen (personal communication). \item A
version of this result with $n=\frac{C}{\varepsilon^{\beta}}$ for
constants $C$ and $\beta$ is proven \cite{Do} and used there to
prove a version of Theorem \ref{theorem:dolgopyatplus} for the $\G$
action on $L^2(M)$.
\end{enumerate}

\noindent {\bf Proof of Theorem \ref{theorem:dolgopyatplus}.}  For
now, assume the closure $K$ of $\G$ in $\Isom(M,g)$ is connected.
The fact that $K$ is compact implies  that for any unitary
representation of $K$ on a Hilbert space $\fh$ which does not
contain the trivial representation and any unit norm vector
$v{\in}\fh$, there is $k{\in}K$ such that
$\|v-d\pi(k)v\|>\frac{1}{2}$. Let $V_j$ be an irreducible,
non-trivial $K$ submodule of $\Vect_2(M)$. Then for any unit vector
$v{\in}V_j$ there is $k{\in}K$ such that $\|v-
d\pi(k)v\|_2>\frac{1}{2}$. We note that the Sobolev embedding
theorems imply that $v-d\tilde \pi(k)v$ is a smooth section of
$K{\times}TM{\rightarrow}K{\times}M$ with all first derivatives
controlled by a constant $D$ times $(1+\lambda_j)^{\alpha}$ where
$\alpha=\frac{\dim(M)+4}{4}$. Letting
$\varepsilon=\frac{1}{4D(1+\lambda_j)^{\alpha}}$ and
 choosing $n$ in Proposition \ref{proposition:fromdima} such that
$n=C(\log(\frac{1}{\varepsilon}))^4$, we can find
$\gamma{\in}B(\G,n)$ such that $|v-d\pi(\g)v|_2>\frac{1}{4}$.
Writing $\g=\g_1{\ldots}\g_{l}$ where each $\g_i{\in}S$ and
$l{\leq}n$, this implies that
$$\sum_{i=1}^l|d\pi(\g_{1}{\ldots}\g_{i-1})v-d\pi(\g_1\ldots\g_i)v|_2>\frac{1}{4}.$$
Since $d\pi$ is unitary, there is an $i$ such that
$\|v-d\pi(\g_i){\inv}v\|_2\geq\frac{1}{4n}$. By our choice of $n$,
 it follows that $$\|v-d\pi(\g_i)v
\|\geq\varepsilon_0(\log(1+\lambda_j))^{-4}.$$

To complete the proof if $K$ is not connected, we let $\G'<\G$ be
$K^0{\cap}\G$, which is clearly a normal subgroup of finite index.
We then choose generators  $S$ of $\G$ which contain a set $S'$ of
generators of $\G'$.  The argument above yields the conclusion of
the theorem except on those $V_j$ which are trivial $K_0$ modules
but not trivial $K$ modules.  For these $V_j$, the $\G$ action is
actually an action of the finite group $\G/\G'$ and so the existence
of a gap, independent of $\lambda$, is trivial. \qed


\subsection{Lower bounds on $d_1$}
\label{subsection:dq}

In this subsection we prove:

\begin{theorem}
\label{theorem:lowerdq} There exist $\varepsilon_0>0$ and a
non-negative integer $\alpha$ such that for each $V_j$ which is a
non-trivial $\G$-module and every $v{\in}V^s_j$ such that $v \perp
\ker(d_1)$ we have
$$\|d_1(v)\|_2{\geq}\frac{\varepsilon_0}{\lambda_j^{\alpha}}\|v\|_2.$$
\end{theorem}

This is considerably more involved than Theorem
\ref{theorem:dolgopyatplus}.  We begin by noting that
$d_1:V^s{\rightarrow}V^r$ and write $d_1^i:V^s{\rightarrow}V$ for
the components.  We actually prove the stronger statement that

$$\|d_1^i(v)\|_2{\geq}\frac{\varepsilon_0}{\lambda_j^{\alpha}}\|v\|_2$$

\noindent for any $v \perp \ker(d_1^i)$. We write $d_1^i(v_1,\ldots,
v_s)=\sum_{k=1}^s\sum_j d\pi(\gamma_{j,k}^i)v_k$ where
$\gamma_{j,k}^i$ are sub-words of the $i$-th relator of $\Gamma$. To
simplify notation, we suppress the $1$ for the remainder of this
subsection and write $d^i=d_1^i$. We further rewrite each $d^i$ as
the composition of two operators $\hat d^i:V^s{\rightarrow}V^s$
defined by $\hat d^i=(d^i_j)$ where $d^i_j(v)=\sum_j
d\pi(\gamma_{j,k}^i)v$ and $d^i=S{\circ}d^i_j$ where
$S:V^s{\rightarrow}V$ is just the sum of the coordinates.

Note that $S$ is uniformly bounded below on $\ker(S)^{\perp}$ and
that $\ker(d^i)= \ker(\hat d^i) \oplus (\hat d^i){\inv}(\ker(S))$.
Letting $W=\hat d^i{\inv}(\ker(S)^{\perp})$ it suffices to bound
$\hat d^i$ below on $W$. This can be done by bounding below each
$d^i_j$ on $W$.  Summarizing, to prove Theorem
\ref{theorem:lowerdq} it suffices to prove the following:

\begin{proposition}
\label{proposition:reducedbound} There exist $\epsilon>0$ and
$\alpha>0$ such that:
\begin{equation}
\label{equation:sufficientdq}
\|d^i_j(v)\|\geq\frac{\epsilon}{\lambda_j^{\alpha}}\|v\|
\end{equation}
for any $v{\in}V_j$ not in $\ker(d^i_j)$.
\end{proposition}

\noindent Since $\Gamma$ is finitely presented, there are only
finitely many $d^i_j$, and one not need to worry if $\epsilon$ and
$\alpha$ depend on $i$ and $j$ or not.  The proof of equation
(\ref{equation:sufficientdq}) is a study of averaging operators of
the form $Av = \sum_j \pi(\gamma_{j})v$. Clearly the $d^i_j$ have
this form.  The principal difficulty here is that we are making no
assumption on the group $\Gamma_A$ generated by the $\gamma_j$
associated to a particular $A$.  Since we cannot use any group
property, instead we control $A$ by controlling the individual
$d\pi(\gamma_j)$, or more precisely, elements of the form
$d\pi(\gamma_i{\inv}\gamma_{j})$.  Given Corollary
\ref{corollary:diophantine}, to prove Proposition
\ref{proposition:reducedbound}, it suffices to prove:

\begin{proposition}
\label{proposition:averagingoperatorbounds} Let $V$ be a Hilbert
space, $\Gamma$ a group acting by unitary transformations on $V$
and and $Av = \sum_{j=1}^n d\pi(\gamma_{j})v$. If we know that
$\|v-d\pi(\gamma_i{\inv}\gamma_j)v\|\geq\eta\|v\|$ whenever $v$ is
perpendicular to the subspace where
$w=d\pi(\gamma_i{\inv}\gamma_j)w$ then $\|Av\|\geq\eta\|v\|$ on
$\ker(A)^{\perp}$.
\end{proposition}

The proof depends on the following easy lemma.  Given an operator
$A$ on a Hilbert space $V$, we let $V_1^A$ be the subspace of $V$ on
which $A$ is the identity.

\begin{lemma}
\label{lemma:inductionstep} Let $A,A'$ be operators on a Hilbert
space $V$ such that $A=v-A'v$.  Assume that on
$(\ker(A'){\oplus}V^{A'}_1)^{\perp}$, we have
$\|v-A'v\|\geq\eta\|v\|$ and $\|A'v\|\geq\eta\|v\|$ then
$\|v-Av\|\geq\eta\|v\|$ and $\|Av\|\geq\eta\|v\|$ on
$(\ker(A){\oplus}V^{A}_1)^{\perp}$.
\end{lemma}

\noindent{\bf Proof:} By definition $\ker(A')=V^A_1$ and
$\ker(A)=V^{A'}_1$ so it suffices to work on the complement to the
sum of these two subspaces.  The proposition then follows from the
equalities $Av=v-A'v$ and $v-Av=A'v$. \qed

\noindent{\bf Proof of Proposition
\ref{proposition:averagingoperatorbounds}} We formally add the
operator $-1$ to the set of elements $d\pi(\gamma_j)$.  Since $-1$
clearly satisfies the required bound, this does not effect our
assumptions.

The proof is an induction on $n$. We first note that the desired
bound for $A$ is equivalent to the same bound for
$d\pi(\gamma_1{\inv}){\circ}A=v-A'v$ where $A'$ is an operator in
the same form as $A$ but with one less term in the sum. The
induction step is now Lemma \ref{lemma:inductionstep} and the base
case is exactly the hypothesis that
$\|v-d\pi(\gamma_i{\inv}\gamma_j)v\|\geq\eta\|v\|$. \qed


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

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

We recall from Section \ref{subsection:theorem1.1}, that the space
of vector fields splits as a Hilbertian 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 and consists of $C^{\infty}$ vector
fields. 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
$$\|v_j-d\pi(\g)v_j\|_2{\geq}\varepsilon_0\lambda_j^{-\alpha}\|v_j\|_2.$$
\end{enumerate}
\end{criterion}
\noindent{\bf Remarks:}\begin{enumerate} \item 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 even provides tame estimates on the size of the
solution in terms of the size of the equation. \item It is
possible to replace the constant
$\varepsilon_0\lambda_j^{-\alpha}$ in $(2)$ with any sequence
$c_j{\inv}$ such that if $\sum_{j=1}^{\infty}a_j \lambda_j^n$
converges for every integer $n$ then $\sum_{j=1}^{\infty}a_j c_j
\lambda_j^n$ converges for every integer $n$.  This is weaker than
what is stated above, but is not relevant to our applications. The
estimate above is analogous to the classical condition of ``small
divisors."

\end{enumerate}

Recall that if we view a vector field on $M$ as a sequence
$\{v_j\}$ where $v_j{\in}V_j$, then smoothness is equivalent to
having the $L^2$ norms $\|v_j\|_2$ decay faster than any
polynomial as a function of $\lambda_j$. So smoothness is
equivalent to $\sum_j{\lambda_j^n}{\|v_j\|_2}<\infty$ for all
positive integers $n$.

\noindent {\bf Proof of Criterion \ref{criterion:vanishing}.}
Given a cocycle $c:\G{\rightarrow}\Vect^{\infty}(M)$, we can write
$c=\{c_j\}$ where $c_j:\G{\rightarrow}V_j$.  Since $H^1(\G,V_j)=0$
for every $j$, it follows that there is a vector $v_j$ such that
$c_j(\g)=v_j-\pi_j(\g)v_j$ for each $j$ and every $\g{\in}\G$. We
now need only see that $\{v_j\}$ represents an element of
$\Vect^{\infty}(M)$.  First note that if $V_j$ is a trivial $\G$
module, then $c_j=0$ and so we can assume $v_j=0$.  If $V_j$ is a
non-trivial $\G$ module then
$\|c_j(\g_i)\|_2=\|v_j-\pi_j(\g)v_j\|_2{\geq}\varepsilon\lambda_j^{-\alpha}\|v_j\|_2$.
Since $\sum\lambda_j^n\|c_j(\g_i)\|$ is finite for all $n$, it
follows that so is $\sum\lambda_j^n\lambda_j^{-\alpha}\|v_j\|_2$
for every $n$, which suffices to show that $v=\{v_j\}$ is smooth
and even to prove a tame estimate
$\|v_j\|_{2,k}{\leq}C_k\|c_j\|_{2,k+\alpha}$. \qed

For all of our applications, Criterion
\ref{criterion:vanishing}$(2)$ can be verified by three methods. The
first is to deduce it directly from Corollary
\ref{corollary:diophantine}. To see that this works, one needs to
note that Proposition \ref{proposition:structure} does not actually
depend on vanishing of $H^1(\Gamma, \Vect^{\infty}(M))$ but only on
vanishing of $H^1(\Gamma,\mathfrak k)$ which is immediate from
$(1)$. The second is to use Theorem \ref{theorem:dolgopyatplus}
above which yields the desired estimate. The third is to use instead
a deep arithmetic result of Clozel, which implies condition $(2)$
with $\alpha=0$, \cite{Cl1}. As noted above, $(2)$ is trivial when
$\G$ is finite and has only has finitely many conjugacy classes of
irreducible representations. We briefly explain how to prove a
version of Criterion \ref{criterion:vanishing}$(2)$ with $\alpha=0$
from work of Clozel in \cite{Cl1}. Clozel's work uses a great deal
of information from the theory of automorphic forms and
representation theory, so though the resulting estimate is stronger,
this approach seems less satisfactory. Theorem \cite[Theorem
3.1]{Cl1} 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 so-called automorphic
spectrum of $G$. The exact statement of Theorem \cite[Theorem
3.1]{Cl1} is that the automorphic spectrum of $G$ is outside some
neighborhood of the trivial representation of $G$ in the unitary
dual.  This implies that the  $\G$ representations we consider are
outside a neighborhood of the identity by elementary properties of
induction of representations.

We conclude this section by verifying that Criterion
\ref{criterion:vanishing} is satisfied in the context of Theorems
\ref{theorem:isomrigid}, \ref{theorem:irredlattices} and
\ref{theorem:clozel}, completing the proofs of those theorems.

\noindent {\bf Proof of Theorem \ref{theorem:isomrigid}.}  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}.  In fact Criterion \ref{criterion:vanishing}$(2)$ with
$\alpha=0$ for all unitary representations of $\G$ is Kazhdan's
original definition of property $(T)$. Criterion
\ref{criterion:vanishing}$(1)$ follows from this by a result of
Guichardet \cite{Gu}. \qed

\noindent {\bf Proof of Theorem \ref{theorem:irredlattices}.}
These lattices satisfy the hypotheses of \cite[Introduction,
Theorem (3)]{Ma}.  This implies that the connected component of
$\pi(\G)$ is semisimple and that $H^1(\G,V)=0$ for all finite
dimensional representations of $\G$. Theorem
\ref{theorem:irredlattices} then follows from Theorem
\ref{theorem:implicit}, Criterion \ref{criterion:vanishing} and
Theorem \ref{theorem:dolgopyatplus}. \qed

\noindent {\bf Proof of Theorem \ref{theorem:clozel}.} In this
case we are assuming that the closure of $\pi(K)$ is a semisimple
compact group.  It remains to verify condition $(1)$ of Criterion
\ref{criterion:vanishing}. This is a special case of \cite[Theorem
3.2 and 3.5]{Cl1}. \qed

\section{Further results}
\label{section:further}

The purpose of this section is to discuss some further
consequences of the methods presented in this paper.  The first
subsection discuss local rigidity results for isometric actions of
more general irreducible lattices in products of locally compact
groups.  The second subsection describes variants on Theorems
\ref{theorem:implicit} and \ref{theorem:affine} for maps to tame
Lie groups other than $\Diff^{\infty}(M)$.  In the final
subsection, we recall from \cite{Cl1}, the definition of the class
of lattices to which Theorem \ref{theorem:clozel} applies.

\subsection{More general results on irreducible lattices}
\label{subsection:irredlattices}

In this subsection, we describe the generalizations of Theorem
\ref{theorem:irredlattices} that were alluded to in the
introduction.  The first concerns so-called $S$-arithmetic
lattices.  We let $A$ be a finite index set and for each
$\alpha{\in}A$, let $k_{\alpha}$ be a local field and
$\Ga_{\alpha}$ an algebraic group over $k_{\alpha}$.  We then let
$G=\prod_A\Ga_{\alpha}(k_{\alpha})$.  We call a lattice $\G<G$
irreducible if the projection to each $\Ga_{\alpha}(k_{\alpha})$
is dense.  If $|A|>1$ and $\G$ is irreducible, then $\G$ is {\em
arithmetic} in the sense of \cite{Ma}, see \cite[Chapter VIII]{Ma}
for a proof, discussion and examples.  In this context, we have

\begin{theorem}
\label{theorem:sarithmetic} Let
$G=\prod_A{\Ga_{\alpha}}(k_{\alpha})$ be as in the last paragraph,
$|A|>1$ and $\G<G$ an irreducible lattice.  Then any isometric
action of $\G$ on a compact manifold is $C^{\infty,\infty}$
locally rigid.
\end{theorem}

\noindent The proof of Theorem \ref{theorem:irredlattices} given
above carries through verbatim in this case.  The simplest example
of a new group to whose actions this applies is
$SL(2,\Za[\frac{1}{p}])$ and one can construct many more examples.
We remark here that this result, and Theorem
\ref{theorem:irredlattices}, are most interesting when the action
of $\G$ is defined by a homomorphism of $\Gamma$ to a compact
group with non-trivial connected component.  There are many
examples of this type in the context of both theorems.

We now introduce some definitions to be able to state a more
general theorem for irreducible lattices in groups that are not a
priori algebraic.  Following \cite{Md}, we say a locally compact
group $G$ has {\em few factors} if
\begin{enumerate}
\item Every non-trivial normal subgroup of $G$ is cocompact. \item
There are no non-trivial continuous homomorphisms
$G{\rightarrow}\Ra$. \item Every closed, normal, cocompact
subgroup of $G$ satisfies $(1)$ and $(2)$.
\end{enumerate}
\noindent Examples of groups with {\em few factors} include
topologically simple groups as well as groups
$\Ga_{\alpha}(k_{\alpha})$ above.

Given $G=G_1{\times}{\ldots}{\times}G_k$ of locally compact
groups, we call a lattice $\G$ {\em totally irreducible} if it's
projection to any $G_i$ is dense and it's projection to any proper
sub-product is not discrete.  If $G$ is locally compact and
compactly generated and $\G<G$ is cocompact, then $\G$ is finitely
generated. We require additional constraints on $\G$. Let $X$ be a
(right) Borel fundamental domain for $\G$ in $G$ and define
$\chi:G{\rightarrow}\G$ by the equation $g{\in}\chi(g)X$ for all
$g{\in}G$. For an element $\g{\in}\G$, we denote by $l(\g)$ the
word length of $\G$ with respect to some chosen generating set.
Then $\G$ in $G$ is said to be {\em square integrable} provided:
\begin{enumerate}
\item $\G$ is finitely presented and \item
$\int_Xl(\chi(g{\inv}h)dh<\infty$ for all $g$ in $G$. \item the
trivial representation of $G$ is isolated in $L^2(G/\Gamma)$ (i.e.
$\Gamma$ is weakly cocompact).

\end{enumerate}
\noindent This is slightly stronger than the definition in
\cite{Md} where $\G$ is only assumed finitely generated. We can
now state our most general result for irreducible lattices in
products.

\begin{theorem}
\label{theorem:generalproducts} Let $I$ be a finite index set and
let $G_i$ be a locally compact, compactly generated group with few
factors.  Let $\G<G=\prod_IG_i$ by a totally irreducible, square
integrable, lattice.  Then any isometric $\G$ action on a compact
manifold is $C^{\infty,\infty}$ locally rigid.
\end{theorem}

\noindent The proof requires only a slight modification from the
proof of Theorem \ref{theorem:irredlattices}.  If
$H^1(\G,V_j)\neq0$, then there is an infinite image linear
representation of $\G$.  By \cite[Theorem 2.4]{Md} this implies
that $\G$ is $S$-arithmetic, which contradicts non-vanishing of
cohomology by \cite[Introduction, Theorem $(3)$]{Ma}.  In fact, if
$\pi(\G)$ is infinite, then \cite[Theorem 2.4]{Md} implies that
Theorem \ref{theorem:generalproducts} reduces to Theorem
\ref{theorem:sarithmetic}.  If $\pi(\G)$ is finite, then all that
remains is to verify \ref{criterion:vanishing}$(2)$.  As remarked
earlier, this is trivial.

\subsection{Rigidity in tame subgroups of $\Diff^{\infty}(M)$.}
\label{subsection:subgroups}

In this subsection we discuss other versions of Theorem
\ref{theorem:implicit} and \ref{theorem:affine} for various other
tame Lie groups.   It is possible to present an axiomatic version of
Theorem \ref{theorem:affine} stated entirely in terms of
homomorphisms $\pi$ from a finitely generated group $\G$ into a tame
Frechet Lie group $D$, given some conditions on the exponential map
of $D$ guaranteeing version of Lemmas \ref{lemma:quadraticerror},
\ref{lemma:differencequotient} and \ref{lemma:notaffine}. It seems
quite difficult to axiomatize what is needed prove Theorem
\ref{theorem:implicit} from Theorem \ref{theorem:affine}. In any
case, this type of generality seems to have no real utility as we
know of no natural tame Frechet Lie groups which are not subgroups
of $\Diff^{\infty}(M)$ for some manifold $M$. The most interesting
variant of Theorem \ref{theorem:implicit} that one can prove
concerns volume preserving diffeomorphisms.  In what follow, we let
$\nu$ be the Riemannian volume form on $M$ and write
$\Vect_{\nu}^{\infty}(M)$ for divergence free vector fields and
$\Diff_{\nu}^{\infty}(M)$ for diffeomorphisms preserving volume.

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

\noindent{\bf Remarks:}\begin{enumerate} \item The proof of this
theorem is exactly the proof of Theorem \ref{theorem:implicit}
using the fact that $\Diff_{\nu}^{\infty}(M)$ is a closed tame
subgroup of $\Diff^{\infty}(M)$ with tangent space at the identity
$\Vect_{\nu}^{\infty}(M)$, see \cite[Theorem III.2.5.3]{Ha} and
the preceding text for discussion . \item This result yields
variants of Theorems \ref{theorem:isomrigid},
\ref{theorem:irredlattices}, \ref{theorem:clozel},
\ref{theorem:sarithmetic} and \ref{theorem:generalproducts} with
slightly stronger assumptions and conclusions.  Namely if one
assumes that the perturbation of the action preserves the
Riemannian volume, then the conjugacy will also be a volume
preserving diffeomorphism.  Even in the setting of Theorem
\ref{theorem:isomrigid}, it does not seem straightforward to prove
this from the techniques of \cite{FM2}. \item There is also an
analogous generalization of  Theorem \ref{theorem:affine} to the
setting of volume preserving diffeomorphisms, whose statement we
leave to the interested reader. \item Whether there are other
versions of this result for diffeomorphisms preserving say a
contact or symplectic structure depends on whether the relevant
subgroups of the diffeomorphism groups are tame Lie groups.  This
question appears to be open, see \cite[Problems III.2.5.5 and
III.2.5.6]{Ha}.
\end{enumerate}

One can also pursue other variants concerning tame Frechet groups
that arise as either sections of principal bundles or subgroups of
diffeomorphism groups preserving foliations.  Variants of Theorems
\ref{theorem:affine} are straightforward, but the resulting
cohomological problems seem quite difficult.

%We mention in passing an additional type of tame subgroups of
%$\Diff^{\infty}(M)$. First, if $M$ is a manifold and $\ff$ is a
%smooth foliation of $M$, then the subgroup of $\Diff^{\infty}(M)$
%consisting of diffeomorphisms preserving each leaf $\ff$ is easily
%seen to be a tame Lie subgroup of $\Diff^{\infty}(M)$. This
%observation, together with the variant of Theorem
%\ref{theorem:affine} that one can prove in this setting, may yield
%an approach to proving some variant of \cite[Theorem 2.11]{FM2} by
%the methods of this paper. The result \cite[Theorem 2.11]{FM2} is
%a foliated generalization of Theorem \ref{theorem:isomrigid} that
%is used in \cite{FM3}.  We do not pursue this direction here, as
%we believe there is a simpler approach to the results in
%\cite{FM3} using only Theorem \ref{theorem:affine}, see the remark
%at the end of subsection \ref{subsection:affine}.

%Finally, if $P$ is a $G$ principal bundle over $M$, then the group
%of sections of $P$ is a tame Lie group.  Analogues of our results
%hold in this setting as well, though it is not quite clear what
%the optimal replacements are for the assumptions that the action
%is affine or isometric.  To obtain a variant of Theorem
%\ref{theorem:implicit}, we clearly need to assume that $G$ is
%compact, though this is not necessary for variants of Theorem
%\ref{theorem:affine}.


\subsection{Fundamental groups of Kottwitz varieties}
\label{subsection:clozel}

In this subsection, we briefly describe the class of lattices to
which Theorem \ref{theorem:clozel} applies.  See \cite[Sections
1.2 and 3.1]{Cl1} for more detailed discussion.  Another useful
reference for details of some constructions is \cite[Chapter
10]{M}.

We first recall the basic objects from which we will construct
arithmetic lattices in $SU(p,q)$. Let $F$ be a totally real number
field $F$, and $L$ a totally imaginary quadratic extension of $F$.
We choose a central division algebra $D$ over $L$ of degree $n^2$
for $n{\geq}3$ with an antiinvolution $\tau$ whose restriction to
$L$ is the Galois automorphism of $L$ over $F$.  We can then
define the group $U(F)=\{d{\in}D|d\tau(d)=1_D\}$.  To construct a
lattice in $U(F)$, one chooses a subring ${\mathcal O}$ of $D$
that is a vector space lattice in $D$.  These can be shown to
exist easily, see \cite[Lemma 10.40]{M}.  The lattice is just
$\G_{\mathcal O}=U(F){\cap}{\mathcal O}$.  The key fact
distinguishing the lattices we consider is that we consider only
those arising in $U(F)<D$ and not more general unitary groups $U$
contained in $M_{l{\times}l}(D)$.

To obtain lattices satisfying Theorem \ref{theorem:clozel}, we
need to impose some further restrictions on $D$.  Our main
non-trivial assumption on $D$ is that at any place $\nu$ of $L$
the algebra $D_{\nu}=D{\otimes}_{L}L_{\nu}$ is either isomorphic
to $M_n(L_{\nu})$ or a division algebra.  As remarked in
\cite{Cl1}, where this condition is called condition $(R)$, this
condition is trivial at infinite places and holds automatically in
at least one finite place. Also as remarked in \cite{Cl1}, this
condition may turn out to be unnecessary. Lastly, we need to
assume that $U(F)$ is isomorphic to $U(p,q){\times}K$ where
$p+q=n$ and $K$ is a product of copies of $U(n)$. This is
equivalent to condition $(K)$ in \cite{Cl1}.

Any finite index subgroup of $\Gamma_{\mathcal O}$ is a lattice in
$U(p,q)$.  To apply the results of Clozel, we need to restrict our
attention to lattices that contain congruence subgroups, i.e.
subgroups obtained by reducing modulo a prime of $F$.  The fact
that many groups satisfying these conditions exist, and even many
with $p=1$, is discussed in \cite[Section 3.1]{Cl1}.





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Hurder, Steven Rigidity for Anosov actions of higher rank
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\bibitem[H2]{H} Hurder, Steven Infinitesimal rigidity for hyperbolic actions.
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Guichardet, Alain Sur la cohomologie des groupes topologiques. II.
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%\bibitem[JM]{JM} Johnson, Dennis; Millson, John J. Deformation spaces
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%J.Jost and S.T.Yau, Harmonic maps and superrigidity, in {\it Lectures on Harmonic Maps} by
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%\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
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\bibitem[Ko]{Ko}
Kononenko, A. Infinitesimal rigidity of boundary lattice actions.
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J. Lewis, Infinitesimal rigidity for the action of $SL_n({\mathbb
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Lubotzky, Alexander; Zimmer, Robert J. Variants of Kazhdan's
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Lubotzky, Alexander; Eigenvalues of the Laplacian, the first Betti
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\bibitem[Ma]{Ma}
G.A.Margulis, {\it Discrete subgroups of semisimple Lie groups},
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%\bibitem[Md1]{Md1}
%Monod, Nicolas, Superrigidity for irreducible lattices and
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\bibitem[Mk]{Mk}
Mok, Ngaiming Harmonic forms with values in locally constant
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\bibitem[Md]{Md}
Monod, Nicolas Arithmeticity vs. nonlinearity for irreducible
lattices.  Geom. Dedicata  112  (2005), 225--237.

\bibitem[M1]{M1}
Morris, Dave Real representations of semisimple Lie algebras have
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\bibitem[M2]{M} Morris, Dave Witte {\it An introduction to
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\bibitem[Q]{Q} Qian, Nantian Infinitesimal rigidity of higher rank lattice
actions. Comm. Anal. Geom. 4 (1996), no. 3, 495--524.

\bibitem[Rg1]{Rg}
Raghunathan, M. S. Discrete subgroups of Lie groups. Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 68. Springer-Verlag, New
York-Heidelberg, 1972.

\bibitem[Rg2]{Rg2} Raghunathan, M. S. Arithmetic lattices in semisimple groups.
Proc. Indian Acad. Sci. Math. Sci.  91  (1982), no. 2, 133--138.



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

%\bibitem[S]{S}
%A. Selberg, On discontinuous groups in higher-dimensional
%symmetric spaces. 1960  {\it Contributions to function theory
%(internat. Colloq. Function Theory, Bombay, 1960)} pp. 147--164
%Tata Institute of Fundamental Research, Bombay

%\bibitem[St]{St}
%Stowe, Dennis, The stationary set of a group action. Proc. Amer.
%Math. Soc. 79 (1980), no. 1, 139--146.

%\bibitem[Th]{Th} W.Thurston,
%A generalization of the Reeb stability theorem. {\it Topology} 13
%(1974), 347--352.

%\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[We]{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}