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\author{V.~M.~Manuilov}
\title{On almost representations of groups
$\pi{\times}{\protect\bf\Large Z}$
\footnote{This research was partially supported by
RFBR (grant No 96-01-00276) and by DFG--RFBR (grant No 96-01-00028).}}
\sloppy
\begin{document}
\maketitle
\noindent
Various generalizations of group representations (like almost and
asymptotic ones) are drawing attention due to their applications to
classification theory of $C^*$-algebras and to the Novikov conjecture on
higher signatures and the Baum--Connes conjecture.
We study here almost representations of discrete
groups $\pi\times\bf Z$ which can be viewed as finite dimensional analogs
of Fredholm representations. We give a construction of such almost
representations and show that for some class of discrete groups
(intermediate between commutative and nilpotent groups) these almost
representations provide enough vector bundles over the classifying spaces
$B\pi\times S^1$. In particular it gives a new proof of the well-known
validity of the Novikov conjecture for these groups.
\noindent
I am grateful to W. L\"uck and to A. S. Mishchenko for helpful
discussions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Construction of almost representations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
Let $\pi$ be a discrete finitely presented group and
fix a number $\e>0$. Let also $F\subset\pi$ be
a finite subset containing generators of $\pi$. By $e$ and $0$ we denote
the neutral elements of the groups $\pi$ and ${\bf Z}$ respectively. We
write elements of the group $\bf Z$ addittively and we
denote by $a\in{\bf Z}$ a generator of this group.
By $U_q({\bf C})$ we denote the unitary group acting on a finite
dimensional Hilbert space ${\bf C}^q$.
\noindent
Notion of almost and asymptotic group representations was introduced in
\cite{c-hig,lor,mish-noor}. But in the present paper we are interested in
a smaller class of representations of the groups of the form $\pi\times\bf
Z$. We will see that these almost representations behave like Fredholm
representations \cite{mish}.
\begin{dfn}
{\rm
A map $\sigma:\pi\times{\bf Z}\arr U_q({\bf C})$ is called an {\em
$\e$-almost representation of the group $\pi\times{\bf Z}$ } (with
respect to $F\subset\pi$) if
\begin{enumerate}
\vspace{-\itemsep}
\item
restriction $\sigma|_{\pi\times 0}$ onto the first coordinate is a usual
representation of the group $\pi$ on ${\bf C}^q$,
\vspace{-\itemsep}
\item
$\sigma(g,na)=\sigma(g,0)\sigma(e,a)^n$ for $n\in{\bf Z}$,
\vspace{-\itemsep}
\item
$\norm{\sigma(g,0)\sigma(e,a)-\sigma(e,a)\sigma(g,0)}<\e$ for all $g\in
F$.
\vspace{-\itemsep}
\end{enumerate}
\noindent
A sequence $\sigma_n$ of $\e_n$-almost representations of the group
$\pi\times\bf Z$ is called an {\em asymptotic representation } if $\e_n\to
0$ and if for every $g\in F$ one has
$$
\norm{\sigma_n(g,0)-\sigma_{n+1}(g,0)}<\e_n,\qquad
\norm{\sigma_n(e,a)-\sigma_{n+1}(e,a)}<\e_n.
$$
}
\end{dfn}
\noindent
Let $X$ be a compact metric space with metric $d$.
\begin{dfn}
{\rm
An action $\a$ of $\pi$ on $X$ is called an {\em $\e$-dense finite action}
(with respect to $F$) if the following properties hold:
\begin{enumerate}
\vspace{-\itemsep}
\item
for any $g\in F$ one has $d(\a_g(x),x)<\e$ for all $x\in X$;
\vspace{-\itemsep}
\item
there exists a point $x_0\in X$ such that its orbit $\{\a_g(x_0):g\in\pi\}$
is finite;
\vspace{-\itemsep}
\item
the orbit $\{\a_g(x_0):g\in\pi\}$ is an $\e$-net in $X$, i.e. for any
$x\in X$ one can find $g\in\pi$ such that $d(x,\a_g(x_0))<\e$.
\vspace{-\itemsep}
\end{enumerate}
}
\end{dfn}
\noindent
We now are going to give a construction of almost representations of
the group $\pi\times{\bf Z}$ (with respect to $F$).
\medskip\noindent
Let $U_q(C(X))$ be the the set of $U_q({\bf C})$-valued
continuous functions on $X$ and let
$u\in U_q(C(X))$ be a unitary element. Then there exists a constant
$C$ such that one has
\be\label{boundedness}
\norm{u(x)-u(y)}0$.
\begin{rmk}
{\rm
In the case when $q=1$ (i.e. when the element $u$ defines a vector bundle
over $X\times S^1$ of rank one) every $\e$-almost representation of
$\pi\times{\bf Z}$ with small enough $\e$ generates an asymptotic
representation of this group. This statement can be proven by the method
of \cite{manCop}. Unfortunately this method cannot be applied for the same
result for arbitrary $q$ and it is unknown if {\it any} almost
representation of $\pi\times{\bf Z}$ generates its asymptotic
representation.
}
\end{rmk}
\noindent
To finish this section we give a simple construction of almost
representations of $\pi\times{\bf Z}$ out of representations of $\pi$.
Let $\rho_t$, $t\in S^1$, be a continuous
loop of finitedimensional representations
of $\pi$. Then an almost representation $\sigma$ of
$\pi\times{\bf Z}$ can be
defined by the matrices
$$
\sigma(g)=
\left(\begin{array}{ccccc}
\!\!\rho_\omega(g)\!\!&&&&\\
&\!\!\rho_{\omega^2}(g)\!\!&&&\\
&&\!\!\rho_{\omega^3}(g)\!\!&&\\
&&&\ddots&\\
&&&&\!\!\rho_{\omega^n}(g)\!\!
\end{array}\right),
\qquad
\sigma(a)=
\left(\begin{array}{ccccc}
0&&&&1\\
1&0&&&\\
&1&0&&\\
&&\ddots&\ddots&\\
&&&1&0
\end{array}\right),
$$
where $g\in\pi$ and $\omega=e^{2\pi i/n}\in S^1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Construction of bundles over $B\pi\times S^1$}\label{bundles}
\label{sect2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
We need the construction~\cite{mish-noor} of vector bundles over
classifying spaces out of almost representations. As we are going to
construct bundles over $B\pi\times S^1$ instead of $B\pi$, so we should
explicitly describe the construction.
\noindent
Let $E\pi\times{\bf R}$ be the total space of the universal
$\pi\times{\bf Z}$-bundle over $B\pi\times S^1$ and let $y\in E\pi$,
$s\in{\bf R}$. Consider a trivial vector bundle $E\pi\times{\bf
R}\times H_n$ over $E\pi\times{\bf R}$. Starting from a $\d$-almost
representation $\sigma_n$ of the group $\pi\times{\bf Z}$ (with small
enough $\d$) we are going to
construct for $g\in\pi$, $b\in{\bf Z}$ transition functions
$$
T_{(g,b)}(y,s):\xi_{(y,s)}\arr\xi_{(gy,bs)}
$$
acting as invertible operators on the fibers $\xi_{(y,s)}$ of this
trivial bundle. The only property that we should check is
\be\label{composition}
T_{(h,c)}(gy,bs)\cdot T_{(g,b)}(y,s)=T_{(hg,cb)}(y,s).
\ee
In fact the transition functions will not depend on $s$, so we
will skip $s$ from our denotations. It is easy to see that in fact it is
sufficient to check the property (\ref{composition}) only for $cb=a$,
namely
\bea
T_{(h,a)}(gy)\cdot T_{(g,0)}(y)&=&T_{(hg,a)}(y),\\
T_{(h,0)}(gy)\cdot T_{(g,a)}(y)&=&T_{(hg,a)}(y).
\eea
Put
\be\label{triv}
T_{(g,0)}(y)=\s_n(g).
\ee
The transition functions (\ref{triv}) define a locally flat vector bundle
over $B\pi$. Unlike them the transition functions $T_{(g,a)}(y)$ depend on
$y$. To define these functions we should take into account the simplicial
structure on $E\pi$. Suppose that $E\pi$ is a finitedimensional simplicial
complex. It is sufficient to define the functions $T_{(g,a)}(y)$ on
the fundamental domain $\ov{Y}\in E\pi$.
At first we define $T_{(g,a)}(y)$ on the zero dimensional skeleton of
$\ov{Y}$. Without any loss of generality we can assume that all vertices of
$\ov{Y}$ constitute one orbit of the natural action of $\pi$.
Let $y_0\in\ov{Y}$ be some vertex.
From now on we suppose that the subset $F\subset\pi$ is big enough so that
all other vertices in $\ov{Y}$ can be obtained from $y_0$ by action
of elements from $F$.
Put
$$
T_{(g,a)}(y_0)=\s_n(g)\cdot\ov{u}.
$$
If $y=hy_0$ is another vertex then the formula (\ref{composition}) gives
$$
T_{(g,a)}(hy_0)=T_{(gh,a)}(y_0)\cdot T^{-1}_{(h,0)}(y_0)=
\s_n(gh)\cdot\ov{u}\cdot\s_n^{-1}(h)=
\s_n(g)\cdot\s_n(h)\,\ov{u}\,\s_n^{-1}(h).
$$
Notice that if $h\in F\subset\pi$ then one has
\begin{eqnarray}
\norm{T_{(g,a)}(hy_0)-T_{(g,a)}(y_0)}&=&
\norm{\s_n(g)(\s_n(h)\,\ov{u}\,\s_n^{-1}(h)-\ov{u})}
=\norm{\s_n(h)\,\ov{u}-\ov{u}\,\s_n(h)}\nonumber\\
&=&\norm{\s_n(h)\,\s_n(a)-\s_n(a)\s_n(h)}<2\d
\end{eqnarray}
by proposition \ref{almrepr}.
Consider a simplex $\Delta=(y_{i_0},\ldots,y_{i_m})\subset\ov{Y}$.
Then we have $y_{i_j}=h_jy_0$ for $h_j\in F$ and
$$
\norm{T_{(g,a)}(y_{i_j})-\s_n(g)\,\ov{u}}<2\d
$$
i.e. $T_{(g,a)}(y)$ are close to the unitary group when $\d$ is small
enough.
Henceforth we can extend the functions $T_{(g,a)}(y)$ to the inner points
of the simplex $\Delta$ by linearity. It is easy to check that the
conditions (\ref {composition}) are satisfied. So we have constructed a
vector bundle over $B\pi\times S^1$. This bundle is a representative of an
element of the group $K^0(B\pi\times S^1)\cong K^0(B\pi)\oplus K^1(B\pi)$.
It is easily seen that the first summand is given by the transition
functions (\ref{triv}) and is a locally flat bundle. Further on we will be
more interested in the second summand.
\noindent
Remark that the above construction gives for every $\e$-dense finite
action of a group $\pi$ on $X$ a homomorphism $K^1(X)\arr K^1(B\pi)$ which
maps $U_q({\bf C})$-valued functions (= maps from $X$ into $U_q({\bf C})$
= elements of $K^1(X)$) into the constructed vector bundles over
$B\pi\times S^1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Almost representations and subgroups of finite index}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
Let $\pi'\subset\pi$ be a subgroup of finite index $m=\#(\pi/\pi')$.
Suppose that the universal covering $E\pi$ of the classifying space $B\pi$
is a finitedimensional simplicial complex with a $\pi$-invariant metric
$d$. Denote by $\ov{Y}_\pi$ (resp. $\ov{Y}_{\pi'}$) the fundamental
domain for the action of the group $\pi$ (resp. $\pi'$) on $E\pi$,
$\ov{Y}_\pi\subset\ov{Y}_{\pi'}\subset E\pi$.
Let $p_\pi$ (resp. $p_{\pi'}$) denote the natural projection of the
fundamental domain onto the classifying space of the group $\pi$ (resp.
$\pi'$). We suppose that the metric $d$ is normed by
${\rm diam}(\ov{Y}_\pi)=1$.
The metric $d$ induces in a natural way the metrics on the classifying
spaces $B\pi$ and $B\pi'$. Suppose that we have a map $\ov{f}:
\ov{Y}_{\pi'}\arr\ov{Y}_\pi$ such that it induces a map $f:B\pi'\arr B\pi$
so that we have a commutative diagram
$$
\diagram
\ov{Y}_{\pi'}\dto_{p_{\pi'}}\rto^{\ov{f}}&
\ov{Y}_\pi\dto^{p_\pi}\\
B\pi'\rto^{f}&B\pi.
\enddiagram
$$
We call such map
$\ov{f}:\ov{Y}_{\pi'}\arr\ov{Y}_\pi$ a {\em $k$-contraction\/} if
one has
$$
d(\ov{f}(y_1),\ov{f}(y_2))0$. \q
\end{cor}
Remark that for the free abelian groups almost representations generate
asymptotic representations \cite{manCop}.
\begin{cor}\label{nil}
Let $G$ be a nilpotent connected simply connected
Lie group, and let $\G\subset G$ be its discrete cocompact subgroup.
Suppose that there exists a linear automorphism $A:G\arr G$ such that
\begin{enumerate}
\vspace{-\itemsep}
\item
$A(\G)\subset\G$;
\vspace{-\itemsep}
\item
all eigenvalues $\l_i$ of $A$ satisfy $|\l_i|\geq \l_0>1$.
\vspace{-\itemsep}
\end{enumerate}
Then the map $\phi\otimes\id_{\bf Q}:
{\cal R}_\e(\G\times{\bf Z})\otimes{\bf Q}\arr
K^1(B\G)\otimes{\bf Q}$ is an epimorphism for small enough $\e$.
\end{cor}
\noindent
{\bf Proof.}
Putting $\G_n=A^n(\G)$ we obtain a sequence
$\G=\G_0\supset\G_1\supset\ldots\supset\G_n\supset\ldots$ such that every
$\G_n$ is isomorphic to $\G$ and every $\G_n$ is of finite index in $\G$.
By results of \cite{Maltsev} all classifying spaces $G/\G_n=B\G_n$ are
diffeomorphic to each other and the map
$\ov{f}=A^{-n}:\ov{Y}_{\G_n}\arr\ov{Y}_\G$ satisfies
$$
d(\ov{f}(x),\ov{f}(y))\leq\l_0^{-n} d(x,y)
$$
for all $x,y\in G$, hence it is a $\l_0^{-n}$-contraction. If $C$ is the
global Lipschitz constant for $B\G$ then the inequality
$C\l_0^{-n}<2^{-\dim G}$ is satisfied for big enough $n$.
Then apply the corollary \ref{cor:epi}. \q
\noindent
Remark that a lot of nilpotent torsion-free discrete groups satisfy the
conditions of the corollary \ref{nil}. The simplest examples
besides the free abelian groups are the
discrete Heisenberg groups and the group of integer upper triangular
matrices.
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\end{thebibliography}
%\end{document}
\vspace{2cm}
\noindent
V.~M.~Manuilov\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119899, RUSSIA\\
e-mail: manuilov@mech.math.msu.su
\end{document}