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\date{}
%\author{V.~M.~Manuilov, \ Chao You}
\title{On Almost Representations of Property (T) Groups}
%----------Author 1
\author{V. M. Manuilov}
\address{%
V. M. Manuilov\\Dept. of Mechanics and Mathematics\\ Moscow State University\\
Leninskie Gory, Moscow\\ 119992 Russia\\and\\Dept. of Mathematics\\
Harbin Institute of Technology\\
Harbin, 150001, P.R.C.}
\email{manuilov@mech.math.msu.su}
\thanks{The first named author was partially supported
by RFFI grant 05-01-00923.}
%----------Author 2
\author{Chao You}
\address{Chao You\\Dept. of Mathematics\\
Harbin Institute of Technology\\
Harbin, 150001, P.R.C.}
\email{hityou1982@hotmail.com}
\sloppy
%\parindent=0em
\begin{document}
\maketitle
\begin{abstract}
Property (T) for groups means a dichotomy: a representation either
has an invariant vector or all vectors are far from being
invariant. We show that, under a stronger condition of \.{Z}uk, a
similar dichotomy holds for almost representations as well.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\G$ be a group generated by a symmetric finite set $S$ and
let $\pi:\G \rightarrow \mathcal {U}(\Hp)$ be a unitary
representation of $\G$. Suppose that $\G$ has the property (T) of
Kazhdan (i.e. the trivial representation is isolated in the dual
space of $\G$). We refer to \cite{HV} for basic information about
property (T). It is well known \cite{HRV} that the spectrum of
$\pi(x)=\frac{1}{|S|}\sum_{s\in\G}\pi(s)$ has a gap near 1:
$$
\Sp(\pi(x))\subset[-1,1-c]\cup\{1\},
$$
where $c$ is the Kazhdan constant for $\G$ (with respect to $S$).
In terms of the group $C^*$-algebra, this means that we can apply
a continuous function $f$ such that $f(1)=1$ and $f(t)=0$ for any
$t\in[-1,1-c]$ to $x=\frac{1}{|S|}\sum_{g\in S} g\in C^*(\G)$ to
obtain the canonical projection $p=f(x)\in C^*(\G)$ corresponding
to the trivial representation \cite{V}.
Our aim is to generalize the above property for the case of almost
representations of $\G$. Recall that, for $\varepsilon\geq 0$, an
$\varepsilon$-almost representation $\pi$ of $\G$ (with respect to
the given set $S$ of generators) is the map $\pi:S\to \mathcal
U(\Hp)$ satisfying
$$
\|\pi(s_1s_2)-\pi(s_1)\pi(s_2)\|\leq\varepsilon
$$
for any $s_1,s_2,s_1s_2\in S$ and $\pi(s^{-1})=\pi(s)^*$ for any
$s\in S$. This definition appeared in \cite{CGM} and then (in a
slightly different form) in \cite{MM}. If $\varepsilon=0$ (in the
case, when $\G$ is finitely presented and $S$ is sufficiently big)
then a 0-almost representation obviously generates a genuine
representation of $\G$. It is known that for some applications it
suffices for $\pi$ to be defined on $S$ only instead of the whole
$\G$. Any small perturbation of a genuine representation is an
almost representation, but there exist almost representations that
are far from any genuine representation \cite{voi}. One should
distinguish almost representations from other group `almost'
notions, e.g. quasi-representations, almost actions etc.
\cite{Stern,F-M}, which are completely different.
If we have an asymptotic representation (i.e. a continuous family
of $\varepsilon_t$-almost representations
$(\pi_t)_{t\in[0,\infty)}$ with
$\lim_{t\to\infty}\varepsilon_t=0$) then it follows from the
theory of $C^*$-algebra asymptotic homomorphisms that the spectrum
of $\pi_t(x)$ has a gap for $t$ sufficiently great: there is a
continuous function $\alpha=\alpha(t)>0$ such that
$\lim_{t\to\infty}\alpha(t)=0$ and
$\Sp(\pi_t(x))\subset[-1,1-c+\alpha(t)]\cup[1-\alpha(t),1]$.
Unfortunately, if we are interested in a single almost
representation, it may be impossible to include an almost
representation into an asymptotic one \cite{M}, and we don't know
how to check existence of a spectral gap because there is no nice
formula for the projection $p$.
Nevertheless, there is a condition, which is only slightly
stronger than the property (T) and which provides a gap in
$\Sp(\pi(x))$ for an almost representation $\pi$. The importance
of this condition was discovered by A. \.{Z}uk \cite{Zuk}. Let us
recall his construction.
It is supposed that the neutral element doesn't belong to $S$. A
finite graph $L(S)$ is assigned to the set $S$ of generators as
follows: the set of vertices of $L(S)$ is $S$ and the set $T$ of
edges of $L(S)$ is the set of all pairs $\{(s,s'):s,s',s^{-1}s'\in
S\}$. By including some additional elements in $S$, one can assume
that the graph $L(S)$ is connected \cite{Zuk}. For a vertex $s\in
L(S)$, let $deg(s)$ denote its {\it degree}, i.e. the number of
edges adjacent to $s$. Let $\Delta$ be a discrete Laplace operator
acting on functions $f$ defined on vertices of $L(S)$ by
\be \label{1.4}
\Delta f(s)= f(s)-\frac{1}{deg(s)}\sum_{s'\sim s}f(s'),
\ee
where $s'\sim s$ means that the vertex $s'$ is adjacent to the
vertex $s$. Operator $\D$ is a non-negative, self-adjoint operator
on the (finitedimensional) Hilbert space $l^2(L(S), deg)$ and zero
is a simple eigenvalue of $\D$. Let $\l_1(L(S))$ be the smallest
non-zero eigenvalue of $\D$. We say that a group $\G$ with the
generating set $S$ satisfies the \.{Z}uk's condition if
$\l_1(L(S))>\frac{1}{2}$. One of the main results of \cite{Zuk}
claims that the \.{Z}uk's condition implies property (T) with the
Kazhdan constant
$c=\frac{2}{\sqrt{3}}\left(2-\frac{1}{\lambda_1(L(S))}\right)$. We
appreciate \.{Z}uk's approach because it allows to work with
almost representations as well. The main result of this paper is
the following statement:
\begin{thm}\label{1}
Let $\G$, $S$ satisfy the \.{Z}uk's condition and let $c$ be as
above. There is a continuous function
$\alpha=\alpha(\varepsilon)\geq 0$ such that $\alpha(0)=0$ and
$$
\Sp\Bigl(\frac{1}{|S|}\sum_{s\in S}\pi(s)\Bigr)\subset
[-1,1-c/2+\alpha(\varepsilon)]\cup[1-\alpha(\varepsilon),1]
$$
for any $\varepsilon$-almost representation $\pi$.
\end{thm}
\begin{cor}
For any $\varepsilon$-almost representation $\pi$ there exists an
$(\varepsilon+6|S|\alpha(\varepsilon))$-almost representation
$\pi'$ such that $\|\pi'(s)-\pi(s)\|\leq 3|S|\alpha(\varepsilon)$
for any $s\in S$ and $\pi'=\tau\oplus\sigma$, where $\tau$ is a
trivial representation and $\sigma$ is an
$(\varepsilon+6|S|\alpha(\varepsilon))$-almost representation
satisfying
\be\label{s1}
\Sp\Bigl(\frac{1}{|S|}\sum_{s\in S}\sigma(s)\Bigr)\subset
[-1,1-c/2+(1+3|S|)\alpha(\varepsilon)].
\ee
\end{cor}
\begin{proof}
Let $H\subset \Hp$ be the range of the spectral projection of
$\frac{1}{|S|}\sum_{s\in S}\pi(s)$ corresponding to the set
$[1-c+\alpha(\varepsilon),1]$. Then
$\|\pi(s)\xi-\xi\|\leq|S|\alpha(\varepsilon)\|\xi\|$ for any $s\in
S$ and for any $\xi\in H$ and if we write $\pi(s)$ as a matrix
$\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)$ with
respect to the decomposition $H\oplus H^\perp$ then
$\|B\|\leq|S|\alpha(\varepsilon)$ and
$\|C\|\leq|S|\alpha(\varepsilon)$, hence there exists a unitary
$D'$ such that $\|D'-D\|\leq 2|S|\alpha(\varepsilon)$. Put
$\pi'(s)=\left(\begin{smallmatrix}1&0\\0&D'\end{smallmatrix}\right)$.
Then $\|\pi'(s)-\pi(s)\|\leq 3|S|\alpha(\varepsilon)$ and $\pi'$
is obviously an $(\varepsilon+6|S|\alpha(\varepsilon))$-almost
representation, which is trivial on $H$. Hence $H^\perp$ is
$\pi(s)$-invariant for all $s\in S$. Since the restriction of
$\frac{1}{|S|}\sum_{s\in S}\pi(s)$ onto $H^\perp$ satisfies
$\bigl(\frac{1}{|S|}\sum_{s\in S}\pi(s)\bigr)|_{H^\perp}\leq
1-c/2+\alpha(\varepsilon)$, we get (\ref{s1}).
\end{proof}
The remaining part of the paper is devoted to the proof of Theorem
\ref{1}. The proof follows the proof of \.{Z}uk for genuine
representations, but has additional argument because relations for
almost representations do not hold exactly, but only
approximately. It will be seen from the proof that one can take
$\alpha(\varepsilon)=O(\varepsilon^{2/5})$ in Theorem \ref{1}.
\section{Proof of the theorem}
The following Hilbert spaces and operators are defined exactly as
in \cite{Zuk}: It doesn't matter that $\pi$ is not a
representation here.
\begin{dfn}[\cite{Zuk}]
{\rm For $r=0,1 \text{ and } 2$ let $C^r$ be the Hilbert spaces
defined as follows:
\begin{eqnarray*}
C^0\!\!\!&=&\!\!\!\{u:u \in \Hp\};\ \langle u,w
\rangle_{C^0}=\langle u,w
\rangle_{\Hp}|T| \text{ for } u,w\in C^0;\\
C^1\!\!\!&=&\!\!\!\{f:S \rightarrow \Hp:f(s^{-1})=-\pi(s^{-1})f(s)
\text{ for all } s\in S \};\ \langle f,g \rangle_{C^1}=\sum_{s\in
S }\langle
f(s),g(s)\rangle_{\Hp}n(s);\\
C^2\!\!\!&=&\!\!\!\{g:T\rightarrow \Hp\};\ \langle f,g
\rangle_{C^2}=\sum_{t\in T }\langle f(t),g(t)\rangle_{\Hp},
\end{eqnarray*}
where $n(s)=\#\{s'\in S:(s,s')\in T\}$. }
\end{dfn}
Since the graph $L(S)$ is connected, $n(s)>0$ for every $s\in S$
and $n(s)=deg(s)$. Moreover, it is easy to see that
$n(s)=n(s^{-1})$ and $\sum_{s\in S}n(s)=|T|$.
\begin{dfn}[\cite{Zuk}]
{\rm Let us define linear operators $d_1:C^0\rightarrow C^1$ and
$d_2:C^1\rightarrow C^2$ as follows:
$$
d_1u(s)=\pi(s)u-u,\text{ for all } u\in C^0;
$$
$$
d_2f((s,s'))=f(s)-f(s')+\pi(s)f(s^{-1}s'), \text{ for all } f\in
C^1.
$$
}
\end{dfn}
\begin{lem}[\cite{Zuk}]
One has $d^*_1f=-2\sum_{s\in S}f(s)\frac{n(s)}{|T|}$ for any $f\in
C^1$ and $\norm{d^*_1}_{C^1\rightarrow C^0}\leq 2$.
\end{lem}
In \cite{Zuk} it is shown that $d_2d_1=0$ for any unitary
representation. However, if $\pi$ is only an almost representation
then this doesn't hold any more. One can only show that this
composition is small.
\begin{lem}\label{5}
For any $u\in C^0$ and $(s,s')\in T$ one has \be \label{2.1}
\norm{d_2d_1u((s,s'))}_{\Hp}\leq \e \norm{u}_{\Hp} \ee
\end{lem}
\begin{proof} By the definitions of $d_1$ and $d_2$, we have
\begin{eqnarray*}
\norm{d_2d_1u((s,s'))}_\Hp&=&\norm{d_1u(s)-d_1u(s')+\pi(s)d_1u(s^{-1}s')}_\Hp\\
&=&\norm{(\pi(s)u-u)-(\pi(s')u-u)+\pi(s)(\pi(s^{-1}s')u-u)}_\Hp\\
&=&\norm{\pi(s')u-\pi(s)\pi(s^{-1}s')u}_\Hp \leq\e\norm{u}_\Hp,
\end{eqnarray*}
hence we have $\norm{d_2d_1u((s,s'))}_{\Hp}\leq \e \norm{u}_{\Hp}$.
\end{proof}
\begin{cor}
$\|d_2d_1\|_{C^2\to C^0}\leq\e$.
\end{cor}
That's why we have to introduce two more (sub)spaces.
\begin{dfn}
{\rm For any $\beta\geq 0$ set
$$
B^0(\beta)=\{P_\Omega(d^*_1d_1)(u):u\in C^0 \}\subset C^0,\qquad
B^1(\beta)=\overline{\{d_1u:u\in B^0(\beta)\}}\subset C^1,
$$
where $P_\Omega$ is the spectral projection corresponding to
$\Omega=[\beta,+\infty)$.}
\end{dfn}
It is clear that $B^0(\beta)$ and $B^1(\beta)$ are invariant
subspaces for $d^*_1d_1$ and $d_1d^*_1$ respectively.
\begin{prop} \label{tm1}
If there exists $c>0$ and $0<\deltac\langle f, f \rangle_{C^1}
\ee
then, for any $\e$-almost representation $\pi$, either there
exists $u\in C^0$ such that
\be\label{6}
\norm{\pi(s)u-u}_\Hp<\delta\norm{u}_\Hp {\mbox{ \ for\ any\ }}
s\in S
\ee
or
\be \label{3}
\max_{s\in S}\norm{\pi(s)u-u}_\Hp\geq c/2\norm{u}_\Hp
\ee
for every $u\in C^0$.
\end{prop}
\begin{proof}
First, we show that if there is no $u\in C^0$ satisfying (\ref{6})
then $B^0(\frac{\delta^2}{|T|})=C^0$. If this is not true then
there exists a non-zero vector $u^\bot$ orthogonal to
$B^0(\frac{\delta^2}{|T|})$. Since $\norm
{d^*_1d_1u^\bot}_{C^0}<\frac{\delta^2}{|T|}\norm{u^\bot}_{C^0}$,
we have $\langle d_1u^\bot,d_1u^\bot \rangle_{C^1}=\langle
u^\bot,d^*_1d_1u^\bot\rangle_{C^0}\leq\norm
{u^\bot}_{C^0}\norm{d^*_1d_1u^\bot}_{C^0}<
\frac{\delta^2}{|T|}\norm{u^\bot}^2_{C^0}$, which implies that
$\norm{d_1u^\bot}_{C^1}<\frac{\delta}{\sqrt{|T|}}\norm
{u^\bot}_{C^0}$. By definition of $\norm{\cdot}_{C^1}$, it is easy
to see that $\norm{\pi(s)u^\bot-u^\bot}_\Hp\leq
\norm{d_1u^\bot}_{C^1}<\frac{\delta}{\sqrt{|T|}}\sqrt{|T|}\norm
{u^\bot}_\Hp=\delta\norm{u^\bot}_\Hp$ for any $s\in S$, which is
in contradiction with the assumption.
Next we prove that (\ref{2.4}) implies that the operator
$d_1d^*_1:B^1(\frac{\delta^2}{|T|})\rightarrow
B^1(\frac{\delta^2}{|T|})$ has a bounded inverse. By (\ref{2.4}),
$d_1d^*_1(B^1(\frac{\delta^2}{|T|}))$ is closed in
$B^1(\frac{\delta^2}{|T|})$. If
$d_1d^*_1(B^1(\frac{\delta^2}{|T|}))$ were different from
$B^1(\frac{\delta^2}{|T|})$, there would exist a non-zero vector
$f \in B^1(\frac{\delta^2}{|T|})$ orthogonal to
$d_1d^*_1(B^1(\frac{\delta^2}{|T|}))$. Then we would have, by
(\ref{2.4}),
$$0=\langle f, d_1d^*_1(f)\rangle_{C^1}> c\langle f, f\rangle_{C^1}$$ which is a contradiction.
Thus $d_1d^*_1:B^1(\frac{\delta^2}{|T|})\rightarrow
B^1(\frac{\delta^2}{|T|})$ has a bounded inverse
$(d_1d^*_1)^{-1}:B^1(\frac{\delta^2}{|T|})\rightarrow
B^1(\frac{\delta^2}{|T|})$ and
$\norm{(d_1d^*_1)^{-1}}_{B^1(\frac{\delta^2}{|T|})\rightarrow
B^1(\frac{\delta^2}{|T|})}\leq c^{-1}$.
Now suppose that neither (\ref{6}) nor (\ref{3}) holds. Then there
is some $\gamma