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\date{}
\author{V.~M.~Manuilov, \ A.~S.~Mishchenko}
\title{Almost, asymptotic and Fredholm representations of discrete groups}
\sloppy
\parindent=0em
\begin{document}
\maketitle
% \begin{abstract}
%To be inserted
% \end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The present paper is a survey of the recent results of the authors
obtained in the framework of the project on almost and asymptotic
group representations partially supported by INTAS (grant No 96-1099).
The study of the so-called almost representations of discrete
groups goes back to Halmos \cite{Ha} and Voiculescu \cite{voi}.
Halmos posed the following problem: suppose that two (unitary or
selfadjoint) matrices almost commute; is it possible to find a
close pair of matrices such that they exactly commute. If one assumes that
the dimension of matrices is bounded then the answer is obviously
positive. If any finite dimension is admitted then the answer
is positive in the case of selfadjoint matrices (a difficult theorem
proved by Lin \cite{Lin}) and negative in the case of unitary matrices.
Namely, Voiculescu \cite{voi} showed that there exists an
asymptotically commuting sequence of matrices without exactly commuting
approximants.
From the point of view of the group representation theory a pair of
commuting matrices can be understood as a representation of the free
abelian group on two generators and the Voiculescu's pair can be
considered as a generalization of such representation. So a notion of a
group almost representation arizes. One of the most striking application
of this generalization was considered for the first time in the paper of
Connes, Gromov and Moscovici \cite{CGM}. They considered a class of almost
flat vector bundles on a manifold $M$, i.e. the
class of elements $\alpha \in K(M)$ such that for any $\e>0$ $\alpha$ can
be written as a difference $\xi -\eta$ of two vector bundles with
almost flat connections, i.e. their curvature does not exceed $\e$.
In this case the parallel transform generates an almost representation of
the fundamental group for every $\e>0$.
It is well known that if $\rho:\pi_1(M)\arr A$
is a unitary representation of the fundamental group $\pi_1(M)$
of a manifold $M$ into a $C^*$-algebra $A$ then one has the generalized
Hirzebruch formula
$$
{\rm sign}_\rho(M)=2^{2k}\left
\in K_0(A)\otimes{\bf Q},
$$
where ${\rm sign}_\rho(M)\in K_0(A)$ is a generalized signature of the
manifold $M$, which is homotopy invariant, $L(M)$ is the Hirzebruch class
of the manifold $M$, $[M]\in H_{4k}(M)$ is its fundamental class,
$\dim M=4k$, $f_M:M\arr B\pi$ is a characteristic mapping and
${\rm ch}_\rho(\xi)\in K^0_A(B\pi)$ is the Chern character of the
canonical $A$-bundle over $B\pi$ generated by a representation
$\rho$.
In the case when $\rho$ is a Fredhlm representation, or more generally, a
representation into the Calkin algebra ${\cal Q}$, this formula gives a
relation between integer-valued invariants because $K_0({\cal Q})=0$ and
$K_1({\cal Q})={\bf Z}$. Another example is provided by the so-called
asymptotic representations because asymptotic representations are in fact
genuine representations, but into more compound `asymptotic'
$C^*$-algebras.
We show
that in some sense the asymptotic representations are a particular case of
representations into the Calkin algebra. Namely we embed the suspension
over the `asymptotic' $C^*$-algebra into the Calkin algebra and show that
this embedding induces an isomorphism of the $K$-groups. It means that any
asymptotic representation of a group $\pi$ gives rise to a representation
of the group ${\bf Z}\times\pi$ into the Calkin algebra.
Remember that the Fredholm representations can be viewed as
representations into the Calkin algebra with some additional structure and
for applications to the Novikov conjecture one uses only the part of the
information contained in the Fredholm representations which remains after
forgetting their more subtle structure. As a corollary we obtain that the
vector bundles over the classifying space $B\pi$ which can be obtained by
asymptotic representations of a discrete group $\pi$ can be also obtained
by representations of the group $\pi\times{\bf Z}$ into the Calkin
algebra. We give also a generalization of the notion of the Fredholm
representations and show that the asymptotic representations can be viewed
as asymptotic Fredholm representations. On the other hand we have found an
example of a group such that its asymptotic representations give much less
vector bundles than its representations into the Calkin algebra.
In fact in the case of $B\pi$ being finitedimensional it is unnecessary to
have a whole asymptotic representation of $\pi$ in order to construct the
corresponding vector bundle. We describe here a construction, which allows
to obtain such almost flat vector bundle out of a single $\e$-almost
representation for some small enough but fixed $\e>0$. We also give a
formula for the first Chern class $c_1(\xi)$ of an almost flat vector
bundle $\xi$ over a manifold $M$ defined by an almost representation
$\sigma$ in terms of the combinatorial structure of $M$ and of $\s$.
The possibility to construct almost flat vector bundles over finite
classifying spaces means that from the point of view of $K$-theory
a single almost representation contains as much information as the
whole asymptotic representation. Hence it is interesting to know for what
groups an almost representation can be extended to an asymptotic one. We
call a group asymptotically stable if any $\e$-almost representation with
small enough $\e>0$ extends to an asymptotic one. In terms of connections
on vector bundles this property means that starting from any almost flat
connection we can continuously infinitely reduce the norm of the curvature
by adding trivial vector bundles with trivial connections on them and that
all intermediate connections are almost flat too. A lot of nice groups
are asymptotically stable but we are able to produce an example of a group
without this property. It should be noted that if we would stabilize by
trivial vector bundles but with arbitrary connections then in our example
it is possible to continuously reduce the norm of the curvature in the
class of almost flat connections.
As a special class of almost representations we consider the class of
those almost representations of the groups ${\bf Z}\times\pi$, whose
restrictions to $\pi$ are genuine representations \cite{M-stekl99}.
We show that for a
class of discrete groups intermediate between commutative and nilpotent
groups the set of such almost representations is sufficient to construct
all vector bundles over the classifying spaces $B\pi\times S^1$.
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic definitions}
Let $\G$ be a finitely presented discrete group, and let
$\G=\la F|R\ra=\la g_1,\ldots,g_n|r_1,\ldots,r_k\ra$ be its presentation
with $g_i$ being generators and $r_j=r_j(g_1,\ldots,g_n)$ being relations.
We assume that the set $F=\{g_1,\ldots,g_n\}$ is symmetric, i.e. for every
$g_i$ it contains $g_i^{-1}$ too, and the set $R$ of relations contains
relations of the form $g_ig_i^{-1}$, though we usually will skip these
additional generators and relations.
By $U_\i$ we denote the direct limit of the groups $U_n$
with respect to the natural inclusion $U_n\arr U_{n+1}$ supplied with the
standard operator norm. The unit matrix we denote by $I\in U_\i$.
\begin{dfn}
{\rm
A set of unitaries $\s=\{u_1,\ldots,u_n\}\subset U_\i$ is called an {\it
$\e$-almost representation} of the group $\G$ if after substitution of
$u_i$ istead of $g_i$, $i=1,\ldots,n$, into $r_j$ one has
$$
\norm{r_j(u_1,\ldots,u_n)-I}\leq\e
$$
for all $j=1,\ldots,k$.
}
\end{dfn}
In this case we write $\s(g_i)=u_i$. Remark that this definition depends
on a choice of presentation of the group $\G$, but we will see that this
dependence is not important. Let $\la h_1,\ldots,h_m|s_1,\ldots,s_l\ra$ be
another presentation of $\G$. For an $\e$-almost representation $\s$ with
respect to the first presentation we can define the set of unitaries
$v_1\ldots,v_m\in U_\i$, $v_i=\ov{\s}(h_i)$ putting
$\ov{\s}(h_i)=\s(g_{j_1})\cdot\ldots\cdot\s(g_{j_{n_i}})$,
where $h_i=g_{j_1}\cdot\ldots\cdot g_{j_{n_i}}$.
By the same way starting
from the set $\ov{\s}(h_i)$ we can construct the set $\ov{\ov{\s}}(g_i)$.
\begin{lem}\label{nezavis}
There exist constants $C$ and $D$ (depending on the two presentations)
such that $\ov{\s}$ is a $C\e$-almost representation with respect to the
second presentation of $\G$ and for all $g_i$, $i=1,\ldots,n$, one has
$\norm{\ov{\ov{\s}}(g_i)-\s(g_i)}\leq D\e$.
\end{lem}
{\bf Proof.}
We have to estimate the norms $\norm{s_q(v_1,\ldots,v_m)-I}$,
$q=1,\ldots,l$. To do so notice that every relation $s_q$ can be written
in the form
\begin{equation}\label{kommut}
s_q=a_1^{-1}r_{j_1}^{\ep_1}a_1\cdot\ldots\cdot
a_{m_q}^{-1}r_{j_{m_q}}^{\ep_{m_q}}a_{m_q}
\end{equation}
for some $a_i\in \G$, where $\ep_i=\pm 1$.
Let $M$ be the maximal length of the words
$a_i=a_i(g_1,\ldots,g_n)$ and $a_i^{-1}=a_i^{-1}(g_1,\ldots,g_n)$.
Substitute $u_1,\ldots,u_n$ into these words and
put $b_i'=a_i^{-1}(u_1,\ldots,u_n)\in U_\i$,
$b_i=a_i(u_1,\ldots,u_n)\in U_\i$. Then one has
$$
\norm{b_i'b_i-I}\leq M\e.
$$
It follows from (\ref{kommut}) that
$$
s_q(v_1,\ldots,v_m)=
b_1'r_{j_1}(u_1,\ldots,u_n)b_1\cdot\ldots\cdot
b_{m_q}'r_{j_{m_q}}(u_1,\ldots,u_n)b_{m_q},
$$
but as for every $i$ one has
$$
\norm{b_i'r_{j_i}^{\ep_i}(u_1,\ldots,u_n)b_i-I}\leq
\norm{b_i'b_i-I}+\norm{r_{j_i}^{\ep_i}(u_1,\ldots,u_n)-I}\leq(M+1)\e,
$$
so
$$
\norm{s_q(v_1,\ldots,v_m)-I}\leq m_q(M+1)\e,
$$
which proves the first statement of the Lemma. The second statement is
proved in a similar way. \q
As the number of generators is finite, so the image of every almost
representation lies in finite matrices, $\s\in U_n$ for some $n$. The
minimal such $n$ is called a dimension of $\s$. Usually we will ignore the
remaining infinite unital tail of the matrices $\s(g_i)$ and write
$\s(g_i)\in U_n$ instead of $U_\i$.
The set of all $\e$-almost representations of the group $\G$ we denote by
${\cal R}_\e(\G)$.
The deviation of an almost representation can be measured by the
value
$$
\nnn{\s}=\max_j\norm{r_j(u_1,\ldots,u_n)-I}.
$$
Notice that both these definitions also depend on the choice of a
presentation of $\G$.
\begin{dfn}[{\rm cf. \cite{c-hig}}]
{\rm
A set of norm-continuous unitary paths
$\s_t=\{u_1(t),\ldots,u_n(t)\}\subset U_\i$, $t\in[0,\i)$, is called an
{\it asymptotic representation} of the group $\G$ if $\nnn{\s_t}$ tends to
zero as $t\to\i$.
}
\end{dfn}
Two asymptotic representations $\s^{(1)}_t$ and $\s^{(2)}_t$ of $\G$ are
called {\it equivalent} if for every $k=1,\ldots,n$ one has
$$
\lim_{t\to\i} \norm{u^{(1)}_k(t)-u^{(2)}_k(t)}=0.
$$
Due to the Lemma \ref{nezavis} this definition does not depend on the
choice of a presentation of $\G$. The Grothendieck group of all
equivalence classes of asymptotic representations of the group $\G$ we
denote by ${\cal R}_{asym}(\G)$.
\medskip
Let $F\subset\G$ be a finite subset.
\begin{dfn}
{\rm
A map $\s:F\to U_\i$ is called an $\e$-almost representation
of the group $\G$ with respect
to $F$ if
\begin{equation}\label{2.1.3}
\norm{\s(gh)-\s(g)\s(h)}\leq\e
\end{equation}
for all $g,h\in F$.
}
\end{dfn}
Note that as before almost representations are defined as maps from a
finite subset of $\G$ into the unitary group, but these maps can be
extended to maps defined on the whole $\G$. Similarly to Lemma \ref{nezavis}
any $\e$-almost representation is a $K\e'$-almost representation with
respect to $F$, where the constant $K$ depends only on the presentation of
$\G$ and on $F$.
\medskip
A discrete version of asymptotic representations turns out to be useful
too.
\begin{dfn}\label{opr3}
{\rm
A sequence
$\s_m=\{u_1(m),\ldots,u_n(m)\}\subset U_\i$, $m\in{\bf N}$, of almost
representations is called a
{\it $($discrete$)$ asymptotic representation} of the group $\G$ if
$\nnn{\s_n}$ tends to zero as $n\to\i$ and if for every $k=1,\ldots,n$
one has
$$
\lim_{m\to\i}\norm{u_k(m+1)-u_k(m)}=0.
$$
}
\end{dfn}
Similarly we have
\begin{dfn}
{\rm
A sequence
$\{\s_m\}$, $m\in{\bf N}$, of $\e_m$-almost
representations is called a
{\it $($discrete$)$ asymptotic representation} of the group $\G$
with respect to $F\subset\G$ if the sequence $\e_m$ vanishes at infinity
and if
$$
\lim_{m\to\i}\norm{\s_{m+1}(g)-u_m(g)}=0
$$
for all $g\in F$.
}
\end{dfn}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Almost representations and vector bundles}\label{construction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
\subsection{Construction of a vector bundle}
Now we want to construct a natural homo\-mor\-phism
\begin{equation}\label{2.16}
\phi : {\cal R}_{asym}(\pi)\mapr{}K(B\pi).
\end{equation}
Let
$M$
be a finite
$CW$ complex with a
fundamental group
$\pi$, $\pi_1(M)=\pi$,
and let
$$
f_M:M\mapr{}B\pi,
$$
be the map as defined before.
Let
$\tilde M$
be
the universal covering of
$M$,
and $p:\tilde M\mapr{}M$
the projection.
Let
$\bar M\subset \tilde M$
be
a fundamental domain, that is,
a finite
closed
subcomplex such that
$p(\bar M)=M$.
Put
\begin{equation}\label{2.163}
F=\{g\in\pi:g(\bar M)\cap\bar M\neq\emptyset\}.
\end{equation}
One can
construct elements
$\alpha$
of the group
$K(B\pi)$
using their restrictions to the
spaces like
$M$.
More precisely, consider the category
${\cal B}\pi$
whose
objects
are
$CW$ complexes
$M$
with fundamental group
$\pi$,
and morphisms are
the homotopy classes of maps
$g:M_1\mapr{}M_2$
such that
the
diagram
$$
\matrix{
M_1 & \mapr{f_{M_1}} & B\pi
\cr
\mapd{g} & & \mapd{=} \cr M_2 & \mapr{f_{M_2}} & B\pi \cr }
$$
commutes.
Consider a natural correspondence
$\alpha$
which associates any space
$M\in {\cal B}\pi$
to a vector bundle
$\alpha(M)\in K(M)$
such
that for any morphism
$g:M_1\mapr{}M_2$
one
has
$$
\alpha(M_1)=g^*(\alpha(M_2)).
$$
Then it is clear that there is a
one to one correspondence between
the set
$\{\alpha\}$
and the set of vector
bundles over
$B\pi$.
For the sake of simplicity, one can restrict only to finite
$CW$ complexes
$M\in{\cal B}\pi$.
To construct a
vector bundle
over
$M$
we proceed as follows.
Consider a trivial
vector
bundle over
$\tilde M$,
$\xi=\tilde M\times {\bf C}^n$,
and an action of the
group
$\pi$
which is compatible with the action on the base
$\tilde M$.
This
action can be described with a matrix function
\begin{equation}\label{2.166}
T_g(x):\xi_x\mapr{}\xi_{gx},\;x\in\tilde M,\;g\in\pi,
\end{equation}
satisfying the condition
\begin{equation}\label{2.167}
T_g(hx)=T_{gh}(x)\circ
T_h^{-1}(x),\; x\in\tilde M,\;g,h\in\pi.
\end{equation}
The function
(\ref{2.166}) with the property (\ref{2.167}) is called the
{\em transition function}.
From (\ref{2.167}) one has
\begin{equation}\label{2.168}
T_e(x)=E,
\; x\in\tilde M,
\end{equation}
where
$e\in \pi$
is the neutral element, and
$E$
is the identity matrix.
It is clear that the condition (\ref{2.167}) can be checked on distinct
orbits
$\pi(x_0)$
of the action of
$\pi$
on the space
$\tilde M$.
Moreover,
the function
(\ref{2.166}) determines its value only in one
fixed point
$x_0\in \pi(x_0)$ by
using the condition (\ref{2.168}).
In fact, if the function
(\ref{2.166}) is defined at the point
$x_0\in \pi(x_0)$
for all
$g\in\pi$
and
satisfies the condition (\ref{2.168}), then one can extend the function
$T_g(x)$
to an arbitrary point
$x\in\pi(x_0)$
using the formula
(\ref{2.167}):
\begin{equation}\label{2.169}
T_g(x)=T_{gu}(x_0)\circ
T_u^{-1}(x_0),\; x\in\tilde M,\;g,u\in\pi,\;
x=ux_0.
\end{equation}
If
$x=x_0$,
then in (\ref{2.169}) one has
$u=e$,
and hence
$T_g(x)=T_{ge}(x_0)\circ
E=T_g(x_0)$.
One needs to verify the condition (\ref{2.167}) for
$x=ux_0$
and
$y=hx=hux_0$.
One has
\begin{equation}\label{2.1610}
T_g(hx)=T_g(hux_0)=T_{ghu}(x_0)\circ
T_{hu}^{-1}(x_0),
\end{equation}
$$
T_{gh}(x)=T_{ghu}(x_0)\circ
T_u^{-1}(x_0),
$$
$$
T_h(x)=T_{hu}(x_0)\circ
T_u^{-1}(x_0).
$$
Therefore
$$
\matrix{
T_g(hx) & = & T_{ghu}(x_0)\circ T_{hu}^{-1}(x_0)\hfill\cr
& = &
T_{ghu}(x_0)\circ T_u^{-1}(x_0)\circ (T_u^{-1}(x_0))^{-1}
\circ
(T_{hu}(x_0))^{-1}\hfill\cr
& = & T_{ghu}(x_0)\circ T_u^{-1}(x_0)\circ (T_{hu}(x_0)\circ
T_u^{-1}(x_0))^{-1}\hfill\cr
& = & T_{gh}(x)\circ T_h(x).\hfill\cr
}
$$
Let
$\bar M\subset \tilde M$
be the
fundamental domain and
let $F$
be as given in (\ref{2.163}).
Then one can consider the restriction
\begin{equation}\label{2.1614}
\bar
T_g(x)=T_g(x),\;g\in F,\;x\in\bar M,gx\in\bar M.
\end{equation}
The function
(\ref{2.1614}) satisfies the condition (\ref{2.167}) for all admissible
$x,g,h$ where
\begin{equation}\label{2.1615}
x,hx,ghx\in\bar M,\;
g,h\in F.
\end{equation}
The function (\ref{2.1614}) with the property (\ref{2.1615}) defines a vector
bundle over
$M$
as well, and will be called also
the transition function.
Therefore one can define the transition function only for a fixed point
$x_0\in (\bar M\cap\pi(x_0))$
and for all
$g\in\pi$
such
that
$gx_0\in\bar M$.
If
$x,
gx\in\bar M\cap\pi(x_0)$
then
$x=hx_0$,
and hence
$h,g,gh\in F$.
One can use the formula (\ref{2.167}) to define
the
values of the transition function for all admissible
$x,g$.
Therefore for the
construction of the vector bundle
$\phi(\sigma)$
as a family of the transition
functions, one should start from the zero dimensional skeleton.
Put
$$
\bar T_g(x_0)=\sigma(g)
$$
for
a representative
$x_0$
of each zero dimensional orbit.
To extend the transition
function from zero dimensional skeleton
to simplexes of higher dimension,
one can use the property that the action
of the group
$\pi$
is free and the property
(\ref{2.1.3}) for
$\varepsilon=1$.
Indeed, in the general case one should do the following.
Let us choose representatives $\{a_{\alpha}\}$
in each orbits of the set
$[\tilde M]_0$
of vertices.
Then the set
$[\bar M]_0=[\tilde M]_0\cap\bar
M$
has
the property
that:
$g(\bar M)\cap \bar M\neq\emptyset$
is equivalent to
$g([\bar M]_0)\cap [\bar M]_0\neq\emptyset$. Therefore there are
elements
$g_a\in F$
such that
for any element
$a\in[\bar M]_0$,
there is an
$\alpha_a$
such
that
$$
a=g_a(a_{\alpha_a}).
$$
Consider
a simplex $\sigma=(b_0,b_1,\dots,b_n)\subset\bar M$.
This means
that
$$\{b_0,b_1,\dots,b_n\}\subset [\bar M]_0.$$ Then according to (\ref{2.163})
one has
$$
b_i=g_i(a_{\alpha_i}),\;i=0,1,\dots,n;\; g_i\in F,
$$
and as
before, one has
$$
\matrix{
T_g(b_i) & = &
T_{gg_i}(a_{\alpha_i})\circ
T^{-1}_{g_i}(a_{\alpha_i}) & \cr & =
& \sigma(gg_i)\circ \sigma^{-1}(g_i),
& gg_i\in F. \cr
}
$$
Therefore for any
$i$
$$
\|T_g(b_i)-\sigma(g)\|= \|\sigma(gg_i)\circ\sigma(g^{-1}_i)-\sigma(g)\|<\varepsilon.
$$
Let
$x\in\sigma$
be a point with the barycentric
coordinates
$\lambda_0,\lambda_1,\dots,\lambda_n$,
$$
x=\sum_i \lambda_i b_i.\;\lambda_i\geq 0,\;\sum_i\lambda_i=1.
$$
Then put
$$
T_g(x)=\sum^n_{i=0}\lambda_i T_g(b_i).
$$
Obviously we have $\|T_g(x)-\sigma(g)\|<\e$ for every $x\in \bar{M}$,
hence $T_g(x)$ is invertible whenever $\e<1$.
The further problem is to establish the fact
when two
$\varepsilon$-almost
representations give the same vector bundles over
$B\pi$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{First Chern class of an almost flat vector bundle}\label{sec4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The
definition \ref{opr3} is more suitable for calculating the first Chern class
$$
c_1(\phi(\sigma))\in H^2(B\pi;R).
$$
Namely, the first Chern class can be described as
a two-dimensi\-o\-nal cocycle, that is,
as a function defined on the family of two dimensional cells of the space
$B\pi$.
Let
$X$
be a finite simplicial complex with the fundamental group
$\pi$.
The simplicial structure on $X$ induces a finite presentation
of the fundamental group $\pi =\pi_{1}(X)$. Namely, let
$\{x_{\alpha }\}$
be the family of all vertices of the space $X$ and $x_{0}$ ($\alpha =0$) be
a fixed point.
Let $T\subset X$ be a minimal one-dimensional tree which covers all
vertices.
Consider a family of polygonal lines, $\{\gamma_{\alpha }\}\subset T$,
which connect the fixed point $x_{0}$ with the vertices $x_{\alpha }$.
Each one-dimensional (oriented) edge is determined by a pair of vertices
$y_{\alpha\beta }=(x_{\alpha },x_{\beta })$. Denote the family of all
edges by $A=\{y_{\alpha\beta }\}$. Then one can define the map
$$
f:A\mapr{}\pi =\pi_{1}(X,x_{0})
$$
by formula
$$
f(y_{\alpha\beta})=
\gamma_{\alpha }\cdot y_{\alpha\beta}\cdot\gamma_{\beta}^{-1}.
$$
Hence one has the homomorphism
$$
f_{F}:F\langle A \rangle\mapr{}\pi =\pi_{1}(X,x_{0}),
$$
extending the map $f$, where $F\langle A \rangle$ is the free group
generated by $A$. Then the kernel $\ker f_{F}$ is the normal subgroup
generated by the family of elements
\begin{eqnarray*}
S&=&\{y_{\alpha\beta}:(x_{\alpha},x_{\beta})\in T\}
\cup \{s_{\alpha\beta\gamma}=
y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot y_{\gamma\alpha}:\hbox{ where }
\nonumber \\&&
(x_{\alpha},x_{\beta},x_{\gamma}) \hbox{ is a 2-dimensional simplex}\}
\nonumber \\
&=& S_{1}\cup S_{2}.
\end{eqnarray*}
The 2-dimensional cohomology can be determined as a 2-dimensional cocycle
which is a function on 2-dimensional simplices
$s\in S_{2}, \ s=(x_{\alpha},x_{\beta},x_{\gamma})$.
\begin{tm}\label{tm2}
The first Chern class
$c_1(\phi(\sigma))$
can be
de\-s\-cribed as a cocycle
\begin{eqnarray}\label{2.17}
c_1(s)&=&\frac{1}{2\pi i}\left(\hbox{tr}\log\sigma_{n}(s)
-\log\det\sigma_{n}(y_{\alpha\beta})\phantom{\int}\right.
\nonumber\\
&&\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\beta\gamma})-
\log\det\sigma_{n}(y_{\gamma\alpha})\right) \in\b{Z} ,
\end{eqnarray}
where
$s=s_{\alpha\beta\gamma}=(x_{\alpha},x_{\beta},x_{\gamma})\in S_{2}$
runs
through all 2-dimensional cells
and $n$ satisfies the condition $\varepsilon_{n}<\frac{1}{3}$.
\end{tm}
\proof First of all let us show that the left-hand side of
the formula (\ref{2.17}) is a cocycle.
Consider a 3-dimensional simplex
$t=(x_{\alpha},x_{\beta},x_{\gamma},x_{\delta})$.
Then
\begin{eqnarray*}
\partial c_{1}(t)&=&c_{1}(x_{\beta},x_{\gamma},x_{\delta})-
c_{1}(x_{\alpha},x_{\gamma},x_{\delta})+
c_{1}(x_{\alpha},x_{\beta},x_{\delta})-
c_{1}(x_{\alpha},x_{\beta},x_{\gamma})\nonumber \\
&=&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\beta\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\beta})
-\log\det\sigma_{n}(y_{\beta\gamma})\phantom{\int}\right.
\nonumber\\
&&\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\gamma\delta})-
\log\det\sigma_{n}(y_{\delta\beta})\right)\nonumber \\
&-&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\alpha\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\alpha})
-\log\det\sigma_{n}(y_{\alpha\gamma})\phantom{\int}\right.
\nonumber\\
&&\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\gamma\delta})-
\log\det\sigma_{n}(y_{\delta\alpha})\right)\nonumber \\
&+&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\delta}\cdot y_{\delta\alpha})
-\log\det\sigma_{n}(y_{\alpha\beta})\phantom{\int}\right.
\nonumber\\
&&\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\beta\delta})-
\log\det\sigma_{n}(y_{\delta\alpha})\right)\nonumber \\
&-&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot y_{\gamma\alpha})
-\log\det\sigma_{n}(y_{\alpha\beta})\phantom{\int}\right.
\nonumber\\
&&\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\beta\gamma})-
\log\det\sigma_{n}(y_{\gamma\alpha})\right)\nonumber \\
&=&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\beta\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\beta})-
\hbox{tr}\log\sigma_{n}
(y_{\alpha\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\alpha})
\phantom{\int}\right.\nonumber \\&+&\left.\phantom{\int}
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\delta}\cdot y_{\delta\alpha})-
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot y_{\gamma\alpha})
\right)\nonumber \\
&=&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(y_{\beta\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\beta})+
\hbox{tr}\log\sigma_{n}
(y_{\alpha\delta}\cdot y_{\delta\gamma}\cdot y_{\gamma\alpha})
\phantom{\int}\right.\nonumber \\&+&\left.\phantom{\int}
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\delta}\cdot y_{\delta\alpha})-
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot y_{\gamma\alpha})
\right)\nonumber \\
&=&\frac{1}{2\pi i}\left(
\hbox{tr}\log\sigma_{n}
(s_{\beta\gamma\delta})+
\hbox{tr}\log\sigma_{n}
(s_{\alpha\delta\gamma})
\phantom{\int}\right.\nonumber \\&+&\left.\phantom{\int}
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\delta})+
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\gamma})
\right).\nonumber \\
\end{eqnarray*}
Since $\|s_{n}(s_{\alpha\beta\gamma})-1\|<\frac{1}{3}$, one has
$\|s_{n}(s_{\alpha\beta\delta}\cdot s_{\alpha\delta\gamma})-1\|<\frac{2}{3}$.
Hence
\begin{eqnarray*}
\hbox{tr}\log\sigma_{n}
(s_{\alpha\delta\gamma})+
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\delta})=
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\delta}\cdot s_{\alpha\delta\gamma})
\nonumber \\ =
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\delta}\cdot y_{\delta\gamma}\cdot y_{\gamma\alpha})=
\hbox{tr}\log\sigma_{n}
(y_{\beta\delta}\cdot y_{\delta\gamma}\cdot y_{\gamma\alpha}\cdot y_{\alpha\beta}).
\end{eqnarray*}
Therefore
\begin{eqnarray*}
\hbox{tr}\log\sigma_{n}
(s_{\beta\gamma\delta})-
\hbox{tr}\log\sigma_{n}
(s_{\alpha\gamma\delta})+
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\delta})
\nonumber \\=
\hbox{tr}\log\sigma_{n}
(y_{\beta\gamma}\cdot y_{\gamma\delta}\cdot y_{\delta\beta})+
\hbox{tr}\log\sigma_{n}
(y_{\beta\delta}\cdot y_{\delta\gamma}\cdot
y_{\gamma\alpha}\cdot y_{\alpha\beta})
\nonumber \\ =
\hbox{tr}\log\sigma_{n}
(y_{\beta\gamma}\cdot
y_{\gamma\alpha}\cdot y_{\alpha\beta})=
\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot
y_{\gamma\alpha})
\nonumber \\ =
\hbox{tr}\log\sigma_{n}
(s_{\alpha\beta\gamma}).
\end{eqnarray*}
Thus
$$
\partial c_{1}(t) =0.
$$
The definition of the right-hand side of the formula (\ref{2.17})
depends of the choice of the value $\log\det\sigma_{n}(y_{\alpha\beta})$.
If one change the value of $\log\det\sigma_{n}(y_{\alpha\beta})$ to
another one:
$$
\log\det\sigma_{n}(y_{\alpha\beta})\longmapsto
\log\det\sigma_{n}(y_{\alpha\beta})+2\pi i m_{\alpha\beta}, \quad
m_{\alpha\beta}\in\b{Z},
$$
the value of $c_{1}(s_{\alpha\beta\gamma})$ will change by the summand
$$
c_{1}(s_{\alpha\beta\gamma})\longmapsto
c_{1}(s_{\alpha\beta\gamma})+
m_{\alpha\beta}+m_{\beta\gamma}+m_{\gamma\alpha}=
c_{1}(s_{\alpha\beta\gamma}+\partial m(s_{\alpha\beta\gamma}).
$$
Therefore the cohomology class defined by the formula (\ref{2.17})
does not depend of the choice of the value of
$\log\det\sigma_{n}(y_{\alpha\beta})$.
Now we shall prove that the cohomology class (\ref{2.17}) coincides
with the first Chern class of the bundle $\phi(\sigma)$.
The bundle $\phi(\sigma)$ is determined by the family of transition functions
$\phi^{n}_{\alpha\beta}(x)$, $x\in U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$,
for the atlas of charts
$U_{\alpha}=\hbox{star}(x_{\alpha})$ such that
\begin{equation}\label{eq(1)a17}
\|\phi^{n}_{\alpha\beta}(x)-\sigma_{n}(y_{\alpha\beta})\|<\varepsilon_{n}.
\end{equation}
Using the transition functions $\phi^{n}_{\alpha\beta}(x)$
the first Chern class $c_{1}(\phi(\sigma))$ is defined by formula
\begin{equation}\label{eq(1)a18}
c_{1}(s_{\alpha\beta\gamma})=
\frac{1}{2\pi i}\left(
\log\det\phi^{n}_{\alpha\beta}(x)+
\log\det\phi^{n}_{\beta\gamma}(x)+
\log\det\phi^{n}_{\gamma\alpha}(x)\right).
\end{equation}
The right-hand side of the formula (\ref{eq(1)a18}) is integer and therefore
it does not depend on the point $x \in U_{\alpha\beta\gamma}=
U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$. In fact, modulo integers one has
\begin{eqnarray*}
\log\det\phi^{n}_{\alpha\beta}(x)+
\log\det\phi^{n}_{\beta\gamma}(x)+
\log\det\phi^{n}_{\gamma\alpha}(x)
\nonumber \\=
\log\left(
\det\phi^{n}_{\alpha\beta}(x)+
\det\phi^{n}_{\beta\gamma}(x)+
\det\phi^{n}_{\gamma\alpha}(x)\right)
\nonumber \\=
\log\det\left(
\phi^{n}_{\alpha\beta}(x)+
\phi^{n}_{\beta\gamma}(x)+
\phi^{n}_{\gamma\alpha}(x)\right)=\log(1)=0.
\end{eqnarray*}
We should compare the two integers, that is we should prove that
\begin{eqnarray}\label{eq(1)a20}
\frac{1}{2\pi i}\left(\hbox{tr}\log\sigma_{n}
(y_{\alpha\beta}\cdot y_{\beta\gamma}\cdot y_{\gamma\alpha})
-\log\det\sigma_{n}(y_{\alpha\beta})\phantom{\int}\right.
\nonumber\\
\left.\phantom{\int}-
\log\det\sigma_{n}(y_{\beta\gamma})-
\log\det\sigma_{n}(y_{\gamma\alpha})\right)
\nonumber \\=
\frac{1}{2\pi i}\left(
\log\det\phi^{n}_{\alpha\beta}(x)+
\log\det\phi^{n}_{\beta\gamma}(x)+
\log\det\phi^{n}_{\gamma\alpha}(x)\right).
\end{eqnarray}
By (\ref{eq(1)a17}) one has
$$
T_{\alpha\beta}(x)\phi^{n}_{\alpha\beta}(x)=\sigma_{n}(y_{\alpha\beta}),
$$
where
$$
\|T_{\alpha\beta}(x)-1\|<\varepsilon_{n}.
$$
There are continuous deformations $T^{t}_{\alpha\beta}(x)$ such that
\begin{eqnarray*}
\|T^{t}_{\alpha\beta}(x)-1\|<\varepsilon_{n},
\nonumber \\
T^{0}_{\alpha\beta}(x)\equiv 1, \quad T^{1}_{\alpha\beta}(x)\equiv
T_{\alpha\beta}(x).
\end{eqnarray*}
Then the integer expression
\begin{eqnarray*}
\frac{1}{2\pi i}\left(
\hbox{tr}\log
(T^{t}_{\alpha\beta}(x)\sigma_{n}\phi_{\alpha\beta}(x)\cdot
T^{t}_{\beta\gamma}(x)\sigma_{n}\phi_{\beta\gamma}(x)\cdot
T^{t}_{\gamma\alpha}(x)\sigma_{n}\phi_{\gamma\alpha}(x))\phantom{\int}\right.
\nonumber\\
\left.-\log\det T^{t}_{\alpha\beta}(x)\sigma_{n}(\phi_{\alpha\beta}(x))
\log\det T^{t}_{\beta\gamma}(x)\sigma_{n}(\phi_{\beta\gamma}(x))
\right.\nonumber\\
-\left.\phantom{\int}
\log\det T^{t}_{\gamma\alpha}(x)\sigma_{n}(\phi_{\gamma\alpha}(x))\right)
\end{eqnarray*}
does not depend on parameter $t$.
Therefore the equality (\ref{eq(1)a20}) holds. \q
As a matter of fact, the construction
of the vector bundle $f^*_M(\phi(\sigma)))$
over
a compact manifold with funda\-men\-tal group
$\pi$
depends only on some sufficiently small
$\varepsilon_n$,
and coincides with
the classical one for
$\varepsilon=0$.
The admissible values for
$\varepsilon$
can be roughly estimated by
$\hbox{exp}(-\dim(M))$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extending almost representations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
\subsection{Asymptotic representations of ${\bf Z}\oplus{\bf Z}$. An example}
\label{sec3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consider an example of fundamental group which seems
to be the simplest nontrivial case when there exists an asymptotical
representation giving nontrivial vector bundle over classifying space.
Let $\pi=Z\oplus Z$, and let $a,\;b\;\in \pi$ be generators of the group
such that $ab=ba$.
To construct an asymptotic representation
it suffices to define two continuous matrix-valued functions,
$A(t),\;B(t)\in U(\infty),\quad 0\leq t < =\infty$ such that
$$
\lim_{t\mapr{}\infty}\|A(t)B(t)-B(t)A(t)\|=0
$$
First of all we shall construct a discrete series of pairs
of unitary matrices (the Voiculescu pairs)
$A_n,\;B_n\in U(n)$.
Let $A_n$ be a matrix of cyclic permutation of the orthonormal basis in
${\bf C}^n$:
$$
A_n=\left(\matrix{
0 & 0 & 0 & \cdots & 0 & 1 \cr
1 & 0 & 0 & \cdots & 0 & 0 \cr
0 & 1 & 0 & \cdots & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr
0 & 0 & 0 & \cdots & 1 & 0
}\right),
$$
and let $B_n$ be the diagonal matrix
$$
B_n=\left(\matrix{
\lambda_1 & 0 & 0 & \cdots & 0 & 0 \cr
0 & \lambda_2 & 0 & \cdots & 0 & 0
\cr 0 & 0 & \lambda_3 & \cdots & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots
\cr 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \cr
0 & 0 & 0 & \cdots & 0 & \lambda_n }\right).
$$
Then the commutator has the following form
\begin{eqnarray*}
\lefteqn{A_nB_n-B_nA_n}\nonumber \\
& =\left(\matrix{
0 & 0 & 0 & \cdots & 0 & \lambda_n \cr
\lambda_1 & 0 & 0 & \cdots & 0 & 0 \cr 0 & \lambda_2 & 0 & \cdots & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr
0 & 0 & 0 & \cdots & \lambda_{n-1} & 0
}\right) & \nonumber \\
& -\left(\matrix{
0 & 0 & 0 & \cdots & 0 & \lambda_1
\cr \lambda_2 & 0 & 0 & \cdots & 0 & 0
\cr 0 & \lambda_3 & 0 & \cdots & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr
0 & 0 & 0 & \cdots & \lambda_n & 0
}\right) & \nonumber \\
= & \left(\matrix{
0 & 0 & 0 & \cdots & 0 & \lambda_n-\lambda_1
\cr \lambda_1-\lambda_2 & 0 & 0 & \cdots & 0 & 0
\cr 0 & \lambda_2-\lambda_3 & 0 & \cdots & 0 & 0
\cr \vdots & \vdots & \vdots & & \vdots & \vdots
\cr 0 & 0 & 0 & \cdots & 0 & 0 \cr
0 & 0 & 0 & \cdots & \lambda_{n-1}-\lambda_n & 0 }\right).
\end{eqnarray*}
Remind that the winding number $w(A,B)$ \cite{voi,e-l} of a pair of
unitaries $A$ and $B$ with $\|AB-BA\|<2$ is the winding number of the loop
given by a map $S^1\to {\bf C}\setminus\{0\}$ defined by the formula
$$
[0,1]\ni t\longmapsto {\rm det}(tAB+(1-t)BA).
$$
For our choice of the unitaries $A$ and $B$ one has $w(A,B)=1$ for every
$n\geq 3$.
The next step is to frame the matrices $A_n$ and $B_n$ with additional row
and column:
$$
\tilde A_n=\left(\matrix{
A_n & 0 \cr
0 & 1
}\right)=
\left(\matrix{
0 & 0 & 0 & \cdots & 0 & 1 & 0 \cr
1 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 1 & 0 & \cdots & 0 & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 0 & 0 & \cdots & 1 & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & 0 & 1
}\right).
$$
$$
\tilde B_n=\left(\matrix{
B_n & 0 \cr
0 & 1
}\right)=
\left(\matrix{
\lambda_1 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & \lambda_2 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 0 & \lambda_3 & \cdots & 0 & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & \lambda_n & 0 \cr
0 & 0 & 0 & \cdots & 0 & 0 & 1
}\right).
$$
The commutator for the framed matrices has the form
\begin{eqnarray*}
\tilde A_n\tilde B_n-\tilde B_n\tilde A_n &=&
\left(\matrix{
A_nB_n-B_nA_n & 0 \cr
0 & 1
}\right) \nonumber \\
&= & \left(\matrix{
0 & 0 & 0 & \cdots & 0 & \lambda_n-\lambda_1 & 0\cr
\lambda_1-\lambda_2 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & \lambda_2-\lambda_3 & 0 & \cdots & 0 & 0 & 0\cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & 0 & 0 & \cdots & \lambda_{n-1}-\lambda_n & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & 0 & 1
}\right).
\end{eqnarray*}
So the norms of two commutators coincide, i.e.
$$
\|A_nB_n-B_nA_n\|=\|\tilde A_n\tilde B_n-\tilde B_n\tilde A_n\| =
\max\{|\lambda_1-\lambda_2|, \dots, |\lambda_n-\lambda_1|\}.
$$
Let
\begin{equation}\label{3.9}
\lambda_k=\exp \Bigl(\frac{2\pi ik}{n}\Bigr),\;k=1, \dots, n.
\end{equation}
Then $\lambda_n=1$ and
$$
|\lambda_k-\lambda_{k+1}|\leq \frac{2\pi}{n}.
$$
The next step is to connect the pair $(\tilde A_n,\tilde B_n)$
and the pair $(A_{n+1},B_{n+1})$ by continuous path $(\bar A_t,\bar B_t)$
which satisfies the inequality
\begin{equation}\label{3.11}
\|\bar A_t\bar B_t-\bar B_t\bar A_t\|\leq\frac{2\pi}{n}.
\end{equation}
Put
$$
\bar A_t=
\left(\matrix{
0 & 0 & 0 & \cdots & 0 & \cos(2\pi t) & \sin(2\pi t) \cr
1 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 1 & 0 & \cdots & 0 & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 0 & 0 & \cdots & 1 & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & -\sin(2\pi t) & \cos(2\pi t)
}\right),
$$
$$
\bar B_t=\tilde B_n=
\left(\matrix{
\lambda_1 & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & \lambda_2 & 0 & \cdots & 0 & 0 & 0 \cr
0 & 0 & \lambda_3 & \cdots & 0 & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & 1 & 0 \cr
0 & 0 & 0 & \cdots & 0 & 0 & 1
}\right).
$$
Then one has
\begin{eqnarray*}
\lefteqn{\bar A_t\bar B_t-\bar B_t\bar A_t=}\nonumber \\
& =\left(\matrix{
0 & 0 & 0 & \cdots & 0 & \cos(2\pi t) & \sin(2\pi t)\cr
\lambda_1 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & \lambda_2 & 0 & \cdots & 0 & 0 & 0\cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots\cr
0 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & -\sin(2\pi t) & \cos(2\pi t)
}\right) & \nonumber \\
& -\left(\matrix{
0 & 0 & 0 & \cdots & 0 & \lambda_1\cos(2\pi t) & \lambda_1\sin(2\pi t) \cr
\lambda_2 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & \lambda_3 & 0 & \cdots & 0 & 0 & 0\cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots\cr
0 & 0 & 0 & \cdots & 0 & 0 & 0\cr
0 & 0 & 0 & \cdots & \lambda_n & 0 & 0\cr
0 & 0 & 0 & \cdots & 0 & -\sin(2\pi t) & \cos(2\pi t)
}\right) & \nonumber \\
& =\left(\matrix{
0 & \cdots & 0 & \nu_n\cos(2\pi t) & \nu_n\sin(2\pi t) \cr
\nu_1 & \cdots & 0 & 0 & 0 \cr
0 & \cdots & 0 & 0 & 0 \cr
\vdots & & \vdots & \vdots & \vdots \cr
0 & \cdots & 0 & 0 & 0\cr
0 & \cdots & \nu_{n-1} & 0 & 0 \cr
0 & \cdots & 0 & 0 & 0
}\right),
\end{eqnarray*}
where
$$
\nu_k=\lambda_k-\lambda_{k+1},\; k=1,\dots,n;\; \lambda_{n+1}=\lambda_1.
$$
When $\lambda_k$ are given by (\ref{3.9}) the inequality (\ref{3.11}) holds.
The second part of path connects the pair
$(\bar A_1=A_{n+1},\; \bar B_1=\tilde B_n)$
with the pair $A_{n+1},\; B_{n+1}$ . Let $\bar A_t=A_{n+1}$ be constant and
let $\bar B_t$ be the diagonal matrix, i.e.
$$
\bar B_t=
\left(\matrix{
\lambda_1(t) & 0 & 0 & \cdots & 0 & 0 & 0 \cr
0 & \lambda_2(t) & 0 & \cdots & 0 & 0 & 0 \cr
0 & 0 & \lambda_3(t) & \cdots & 0 & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \cr
0 & 0 & 0 & \cdots & \lambda_{n-1}(t) & 0 & 0 \cr
0 & 0 & 0 & \cdots & 0 & \lambda_n(t) & 0 \cr
0 & 0 & 0 & \cdots & 0 & 0 & \lambda_{n+1}(t)
}\right),
$$
where
\begin{eqnarray*}
&\lambda_k(t)=\exp 2\pi ik((1-t)\frac{1}{n}+t\frac{1}{n+1}),& \;1\leq k\leq n\nonumber\\
&\lambda_{n+1}(t)=1.
\end{eqnarray*}
Therefore when $t=1$, one has
$$
\bar B_1=B_{n+1}.
$$
Thus we have constructed a continuous family of (stable) matrices
$(A_t, B_t)$ such that
$$
\|A_tB_t-B_tA_t\|\leq \frac{2\pi}{[t]},
$$
where $A_n$, $B_n$ coincide with discrete sequence.
Now let us apply the formula (\ref{2.17}) to calculate the Chern class
of the asymptotic representation constructed above.
Since
$$
s=aba^{-1}b^{-1},
$$
and hence
$$
\sigma_n(s)=A_nB_nA_n^{-1}B_n^{-1}.
$$
It is easy to verify that
\begin{eqnarray*}
\lefteqn{A_nB_nA_n^{-1}B_n^{-1}=}\nonumber \\
&=\left(\matrix{
\frac{\lambda_1}{\lambda_2} & 0 & 0 & \cdots & 0 & 0 \cr
0 & \frac{\lambda_2}{\lambda_3} & 0 & \cdots & 0 & 0
\cr 0 & 0 & \frac{\lambda_3}{\lambda_4} & \cdots & 0 & 0 \cr
\vdots & \vdots & \vdots & & \vdots & \vdots
\cr 0 & 0 & 0 & \cdots & \frac{\lambda_{n-1}}{\lambda_n} & 0 \cr
0 & 0 & 0 & \cdots & 0 & \frac{\lambda_n}{\lambda_1}
}\right) & \nonumber \\
&=\left(\matrix{
\exp (\frac{-2\pi i}{n}) & 0 & \cdots & 0 & 0 \cr
0 & \exp (\frac{-2\pi i}{n}) & \cdots & 0 & 0 \cr
0 & 0 & \cdots & 0 & 0 \cr
\vdots & \vdots & & \vdots & \vdots \cr
0 & 0 & \cdots & \exp (\frac{-2\pi i}{n}) & 0 \cr
0 & 0 & \cdots & 0 & \exp (\frac{-2\pi i}{n})
}\right). &
\end{eqnarray*}
Thus
$$
c_1(s)=\hbox{tr}(\log(\sigma(s)))/(2pi) =1.
$$
Consequently, one has
\begin{tm}\label{tm3}
The homomorphism
$$\varphi :{\cal R}_{asym}(Z\oplus Z)\mapr{} K(T^2)$$
is surjective.
\end{tm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic stability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The example above suggests to investigate in general a problem when
an almost representation can be extended to an asymptotic one.
Now we are ready to define a new property of finitely generated groups
which we call {\it asymptotic stability}.
\begin{dfn}
{\rm
A group $\G$ is asymptotically stable if for every $\e>0$
one can find a number $\d(\e)$ (with the property $\d(\e)\to 0$
when $\e\to 0$) such that for every
almost representation $\s\in{\cal R}_\e(\G)$ there exists an asymptotic
representation $\s_t\in{\cal R}_{asym}(\G)$ such that $\s_0=\s$ and
$\nnn{\s_t}\leq \d(\e)$ for all $t\in [0,\i)$.
}
\end{dfn}
Requirement that $\nnn{\s_t}$ should be small enough for {\it all} $t$
is necessary because without it any two almost representations can be
connected by a path in the unitary group, since the latter is path
connected. On the other hand examples show that it would be too
restrictive to demand that $\d(\e)=\e$.
\begin{lem}
Asymptotic stability property does not depend on the choice of a
presentation of the group.
\end{lem}
{\bf Proof } immediately follows from the Lemma \ref{nezavis}. \q
The next theorem describes classes of asymptotically stable discrete
groups.
\begin{thm}
The following groups are asymptotically stable:
\begin{enumerate}
\vspace{-\itemsep}
\item
free groups,
\vspace{-\itemsep}
\item
free products of asymptotically stable groups,
\vspace{-\itemsep}
\item
finite groups,
\vspace{-\itemsep}
\item
abelian groups,
\vspace{-\itemsep}
\item
fundamental groups of two-dimensional oriented manifolds.
\vspace{-\itemsep}
\end{enumerate}
\end{thm}
{\bf Proof.} The first item is obvious --- free groups have no relations,
so every almost representation is a genuine representation.
The same argument works for the second item too.
The third item was proved in \cite{Grov} --- for finite groups there
exists a genuine representation close to every almost representation.
We prove the fourth and the fifth items in the next section.
%\cite{M-izv99,M-timish}. \q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic stability of some classes of groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we prove the asymptotic stability property for
abelian groups and for fundamental groups of oriented two-dimensional
surfaces, cf. \cite{M-izv99,M-timish}.
Let the set $u_1,\ldots,u_k,v_1,\ldots,v_l\in U_n({\bf C})$ be
an $\e$-almost representation of the abelian group $\G={\bf
Z}^k\oplus{\bf Z}_{n_1}\oplus\ldots\oplus{\bf Z}_{n_l}$ generated by free
generators $a_1,\ldots,a_k$ and by torsion generators $b_1,\ldots,b_l$
with $b_j^{n_j}=1$. Then one has $\norm{[u_i,u_j]}\leq\e$,
$\norm{[u_i,v_j]}\leq\e$, $\norm{[v_i,v_j]}\leq\e$ and
$\norm{v_j^{n_j}-{\bf 1} }\leq\e$.
\begin{thm}\label{alm->asym}
There exist constants $\e_0>0$ and $C>0$ such that for any
$\e$-almost representation of the group $\G$ with $\e<\e_0$ there exists
a dimension $N$ and
unitary paths $u_i(t),v_j(t)\in U_N({\bf C})$ such that
\begin{enumerate}
\item
$u_i(0)=u_i\oplus {\bf 1} _{N-n}$, $v_j(0)=v_j\oplus {\bf 1} _{N-n}$,
\vspace{-\itemsep}
\item
all the norms $\norm{[u_i(t),u_j(t)]}$,
$\norm{[u_i(t),v_j(t)]}$, $\norm{[v_j(t),v_i(t)]}$,
$\norm{v_j^{n_j}-{\bf 1} }$ do not exceed $C\e^{(1/4)^{k-1}}$,
\vspace{-\itemsep}
\item
$\norm{[u_i(1),u_j(1)]}\leq\e/2$ and $[u_i(1),v_j(1)]=[v_j(1),v_i(1)]=
v_j^{n_j}-{\bf 1} =0$.
\end{enumerate}
\end{thm}
\noindent
{\bf Proof.}
We start with proof of Theorem \ref{alm->asym} in the case of $k=2$,
$l=0$, i.e. for two almost commuting unitaries $u=u_1$ and $v=u_2$
being the images of two free generators of the group ${\bf Z}\oplus{\bf Z}$.
In the special case of the pair of the Voiculescu matrices it was shown in
\cite{mish-noor} that they generate an asymptotic representation of this
group.
\medskip\noindent
Our proof is based on several technical lemmas. We begin with these
lemmas in the first section and afterwards we will give an explicit
construction of the homotopy $u(t)=u_1(t)$ and $v(t)=u_2(t)$. The idea is
that if one of two almost commuting matrices is diagonal then the other
is almost of a block three-diagonal form (cf. \cite{manFA}) and
afterwards we use a variant of the Berg's technique (cf.~\cite{lor-berg}).
The general case will easily follow from the case of two unitaries and we
deal with it later.
\medskip\noindent
We do not try to get the best possible estimates and the constants surely
can be chosen in a better way.
%\smallskip\noindent
For convenience we assume that all
$\e$, $\d_1$, $\d_2$ etc. take values in the interval $(0,1)$.
\subsection*{Technical lemmas}\label{techn.lemmas}
%\setcounter{equation}{0}
We start with two unitaries $u,v\in U_n({\bf C})$ and fix a basis such that
the matrix $u$ is diagonal with diagonal entries $\l_i=e^{2\pi i\ph_i}$,
$i=1,\ldots,n$,
with ordered values: $\ph_i\in [0,1)$, $\ph_{i+1}\geq\ph_i$.
Let $\alpha$ be such that $|e^{2\pi i\alpha}-1|=\sqrt[4]{\e}$,
then the points $e^{2\pi ik\alpha}$, $k=0,1,\ldots$, divide the unit
circle into equal small arcs
$[e^{2\pi ik\alpha},e^{2\pi i(k+1)\alpha})$
(in fact the last arc $[e^{2\pi iK\alpha},1)$ may be shorter
than others but without loss of generality we can assume that $\e$ is such
that this last arc is of the same length too).
Denote those arcs which contain at least one eigenvalue $\l_i$ by
$\D_k$. Then the number $l$ of such arcs is not bigger then
$2\pi/\sqrt[4]{\e}$ and the following properties hold:
\begin{enumerate}
\item
if $\l_i,\l_j\in\D_k$ then $\v \l_i-\l_j\v <\sqrt[4]{\e}$;
\vspace{-\itemsep}
\item
if $\l_i\in\D_{k-1}$ and $\l_j\in\D_{k+1}$ then $\v \l_i-\l_j\v
\geq\sqrt[4]{\e}$.
\end{enumerate}
\noindent
Denote the orthogonal spectral projections of $u$ corresponding to the
arcs $\D_k$ by $p_k$, $p_1\oplus\ldots\oplus p_l={\bf 1} $. Then the matrix
algebra $M_n$ as a module over itself can be decomposed into a direct sum
corresponding to the above projections, $M_n=\oplus_{k=1}^lp_kM_n$ and we
can represent matrices from $M_n$ as smaller matrices of blocks with
regards to this decomposition: $u=\diag(\{u_k\})$ for $u_k=p_kup_k$ and
$v=(v_{km})$ for $v_{km}=p_kvp_m$.
\medskip\noindent
For a matrix $a=(a_{km})$ decomposed with respect to $\oplus_{k=1}^l
p_kM_n$ put
\be\label{tridiag}
d(a)=\left(\begin{array}{cccc}
a_{11}&a_{12}&&a_{1l}\\
a_{21}&a_{22}&\ddots&\\
&\ddots&\ddots&a_{l-1,l}\\
a_{l1}&&a_{l,l-1}&a_{l,l}
\end{array}\right)
\ee
where empty places are zeroes.
We call matrices of the form (\ref{tridiag}) block three-diagonal (we look
at the corner elements $a_{l1}$ and $a_{1l}$ as at continuations of the
diagonals $(a_{12},\ldots,a_{l-1,l})$ and $(a_{21},\ldots,a_{l,l-1})$
respectively).
\medskip\noindent
We begin by proving that the matrix
$v=(v_{km})$ is ``almost'' block three-diagonal.
\begin{lem}\label{lemma01}
If
\begin{equation}\label{ac1}
\norm{[u,v]}\leq\e
\end{equation}
then
$$
\norm{v-d(v)}\leq4\sqrt{\pi}\sqrt[4]{\e}.
$$
\end{lem}
\noindent
{\bf Proof.}
%\medskip\noindent
Divide once more the spectrum of the operator $u$ into smaller (than
$\D_k$) equal arcs $\ov{\D}_s=[d_s,d_{s+1})$ such that
$|d_s-\ov{\l}_s|=|\ov{\l}_s-d_{s+1}|=\e/2$, where $\ov{\l}_s$ are the
middle points of the arcs $\ov{\D}_s$.
Let $\ov{p}_s$ be the spectral projections of $u$ corresponding to
the arcs $\ov{\D}_s$. Then put
$$
\ov{u}=\sum_s \ov{\l}_s\ov{p}_s.
$$
Obviously $\norm{u-\ov{u}}\leq\e/2$, hence in view of (\ref{ac1})
\be\label{ac.2}
\norm{\ov{u}v-v\ov{u}}\leq \norm{\ov{u}v-uv}+
\norm{uv-vu}+\norm{vu-v\ov{u}}\leq 2\e.
\ee
Let $M_n=\oplus_s\ov{p}_sM_n$ be the decomposition of $M_n$
corresponding to the spectral projections $\ov{p}_s$ of $u$. It is a
sub-decomposition of $\oplus_{k=1}^lp_kM_n$ and the matrix $v$ can be
written also as $v=(w_{ij})$,
$w_{ij}=\ov{p}_i v\ov{p}_j$. The matrix entries $v_{km}$
can be viewed as blocks of elements $w_{ij}$.
Denote by $M$ the number of columns of the matrix $(w_{ij})$. Then one has
\be\label{N}
M\leq\frac{4\pi}{\e}.
\ee
\medskip\noindent
%{\bf Proof.}
Consider the matrix $(\ov{u}v-v\ov{u})(\ov{u}v-v\ov{u})^*$.
From the inequality
(\ref{ac.2}) it follows that the norm of this matrix
is less then $4\e^2$,
hence the norm of any element of this matrix is also less than $4\e^2$.
So, as $\ov{\l}_j$ commute with $w_{ij}$, we obtain for every $i$
\be\label{4}
\norm{\sum_{j=1}^M (\ov{\l}_i-\ov{\l}_j)^2 w_{ij}w^*_{ij}}\leq 4\e^2.
\ee
Let $\ov{\l}_i\in\D_{k_i}$, $\ov{\l}_j\in\D_{k_j}$.
As all the summands in (\ref{4}) are positive, so ignoring some
of them we will not increase the norm of the sum, hence
$$
\norm{\sum\nolimits'_j (\ov{\l}_i-\ov{\l}_j)^2 w_{ij}w^*_{ij}}\leq 4\e^2
$$
where the sum $\sum'$ is taken for those $j$ for which
$\v k_j- k_i\v\geq 2$, i.e. we throw away those $w_{ij}$
for which $(v-d(v))_{ij}=0$. As in the sum $\sum'$ we have
$\v\ov{\l}_k-\ov{\l}_j\v\geq\sqrt[4]{\e}$, so
$$
4\e^2>\norm{\sum\nolimits'_j
(\ov{\l}_i-\ov{\l}_j)^2w_{ij}w^*_{ij}}\geq\sqrt{\e}\norm{\sum\nolimits'_j
w_{ij}w^*_{ij}},
$$
hence
\be\label{3}
\norm{\sum\nolimits'_j w_{ij}w^*_{ij}}\leq 4\e^{3/2}.
\ee
To finish the proof we need the following
\begin{lem}\label{NN}
Let $A$ be a $C^*$-algebra and let
$a=(a_{ij})$, $a_{ij}\in A$ be a $M\times M$ matrix such that
for every $i$ one has $\norm{\sum_j a_{ij}a^*_{ij}}\leq\d_1^2$.
Then $\norm{a}\leq\d_1\sqrt{M}$.
\end{lem}
\noindent
{\bf Proof.}
Take $\xi=(\xi_k)$, $k=1,\ldots, M$, $\xi_i\in M_n$.
Then using the generalized
Cauchi-Schwartz inequality~\cite{lance} we get
\begin{eqnarray*}
\norm{a\xi}^2&=&\norm{\sum_{ijk}\xi^*_j a^*_{ij}a_{ik}\xi_k}
\leq\sum_i\norm{\sum_{kj}\xi_j^* a^*_{ij}a_{ik}\xi_k}\\
&\leq&\sum_i\norm{\sum_k a_{ik}a^*_{ik}}\cdot\norm{\sum_k \xi_k^*\xi_k}
\leq M\d_1^2 \norm{\xi}^2,
\end{eqnarray*}
hence we have $\norm{a\xi}\leq\sqrt{M}\d_1\norm{\xi}$. \q
\noindent
Now in view of (\ref{N}) and (\ref{3}) from Lemma \ref{NN}
(for $\d_1=2\e^{3/4}$) we get
$$
\norm{v-d(v)}\leq\sqrt{M}\ 2\e^{3/4}\leq
\sqrt{4\pi}\e^{-1/2}\,2\e^{3/4}= 4\sqrt{\pi}\sqrt[4]{\e}
$$
which ends the proof of Lemma \ref{lemma01}. \q
\noindent
The idea of chopping the space into spectral projections of one
operator and perturbing the other to be three-diagonal has appeared in
\cite{Dav}. Moreover, K. Davidson informed us that using the
Bhatia-Davis-McIntosh method \cite{Bhat} it is possible to get a better
estimate than in Lemma \ref{lemma01}.
\begin{rmk}\label{lacunas}
{\rm
If the spectrum of the matrix $u$ has a gap of length more
than $\sqrt[4]{\e}$ then without loss of generality we can assume that
this gap has zero as its central point. In that case norm of the elements
$v_{1l}$ and $v_{l1}$ is small too and we have
$$
\norm{v-\left(\begin{array}{cccc}
v_{11}&v_{12}&&\\
v_{21}&v_{22}&\ddots&\\
&\ddots&\ddots&v_{l-1,l}\\
&&v_{l,l-1}&v_{l,l}
\end{array}\right)}\leq 4\sqrt{\pi}\sqrt[4]{\e}.
$$
If there exists another gap of length bigger than $\sqrt[4]{\e}$ in
the spectrum of $u$ (say between $\D_r$ and $\D_{r+1}$) then the matrix
$d(v)$ is close to a matrix decomposable into a direct sum of smaller
three-diagonal matrices because the norm of elements $v_{r+1,r}$ and
$v_{r,r+1}$ is small enough.
}
\end{rmk}
%\medskip\noindent
Put
$$
c=c(t)=e^{\pi it/2}\cos{\pi t/2},\qquad s=s(t)=-ie^{\pi it/2}\sin{\pi
t/2},\qquad t\in [0,1].
$$
Then both paths
$\left(\begin{array}{cc}c&s\\s&c\end{array}\right)$ and
$\left(\begin{array}{cc}\ov{c}&\ov{s}\\\ov{s}&\ov{c}\end{array}\right)$
connect matrices $\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$
and $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ in the group
$U_2({\bf C})$.
For a matrix
\be\label{3diag}
w=\left(\begin{array}{cccc}
v_{11}&v_{12}&&v_{1l}\\
v_{21}&v_{22}&\ddots&\\
&\ddots&\ddots&v_{l-1,l}\\
v_{l1}&&v_{l,l-1}&v_{l,l}
\end{array}\right)
\ee
and for an integer $m$
define a path $w(t)$ by formula %$w(t)=$
%{\small
\be\label{bigmatr}
w(t){=}\!
\left(\!\!
\begin{array}{ccccccccccccc}
v_{11}&v_{12}&&cv_{1l}&&&&&&&&&sv_{1l}\\
v_{21}&v_{22}&\ddots&&&&&&&&&&\\
&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!&&&&&&&&&\\
\ov{c}v_{l1}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}&\ov{s}v_{l1}&&&&&&&&\\
&&&sv_{1l}&v_{11}&v_{12}&&cv_{1l}&&&&&\\
&&&&v_{21}&v_{22}&\ddots&&&&&&\\
&&&&&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!&&&&&\\
&&&&\ov{c}v_{l1}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}&\ov{s}v_{1l}&&&&\\
&&&&&&&sv_{1l}&\ddots&&&&\\
&&&&&&&&&v_{11}&v_{12}&&cv_{1l}\\
&&&&&&&&&v_{21}&v_{22}&\ddots&\\
&&&&&&&&&&\ddots&\ddots&\!\!v_{l{-}1,l}\!\!\\
\ov{s}v_{l1}&&&&&&&&&\ov{c}v_{1l}&&\!\!v_{l,l{-}1}\!\!&v_{l,l}\\
\end{array}\!\!
\right)
\ee
%}
\noindent
where for $t=0$ the matrix $w$ is repeated $m$ times. Let $N=nm$ be the
dimension of $w(t)$. We show that the path $w(t)$ lies close to the
unitary group $U_N({\bf C})$ and that the corresponding estimate does not
depend on the number $m$.
\begin{lem}\label{bigmatrix}
Suppose that $\norm{v-w}\leq\d_2$ and $\norm{[u,w]}\leq\d_3$. Let
$w(t)=\til{w}(t)h(t)$ be the polar decomposition with unitary $\til{w}(t)$.
Then
$$
\norm{w(t)-\til{w}(t)}\leq 100\d_2 \quad{\it and}\quad \norm{[u\oplus
{\bf 1} _{N-n},w(t)]}\leq \d_3+4\sqrt[4]{\e}.
$$
\end{lem}
\noindent
{\bf Proof.}
Notice that the inequality $\norm{v-w}\leq\d_2$ implies
$\norm{{\bf 1} -v^*w}\leq\d_2$.
Let $w=\til{w}h$ be the polar decomposition. Then we have
\be\label{ner-vo1}
\norm{{\bf 1} -v^*w}=\norm{\til{w}^*v-h}\leq\d_2.
\ee
Passing to adjoints we obtain
\be\label{ner-vo2}
\norm{\til{w}^*v-v^*\til{w}}\leq 2\d_2,
\ee
hence from (\ref{ner-vo1}) we get
$$
\norm{\til{w}^*v-{\bf 1} }\leq 2\d_2,
$$
and in view of (\ref{ner-vo1})
$$
\norm{h-{\bf 1} }\leq 3\d_2.
$$
Therefore we get $\norm{h^2-{\bf 1} }\leq (1+3\d_2)^2-1\leq 15\d_2$, hence
$$
\norm{w^*w-{\bf 1} }\leq 15\d_2.
$$
Notice that the matrix $w^*w-{\bf 1} $ is 9-diagonal (or 5-diagonal, if we
think of the upper right and lower left corners of this matrix as of
continuations of the diagonals close to the main diagonal), therefore the
norm of every diagonal is not more than $15\d_2$.
\medskip\noindent
Now look at the matrix $w^*(t)w(t)-{\bf 1} $. It is a 13-diagonal matrix and
it is easily seen that every diagonal of this matrix is a direct sum of
diagonals of the matrix $w^*w-{\bf 1} $ (possibly with coefficients like
$c(t)$ and $s(t)$), hence we get the estimate
$$
\norm{w^*(t)w(t)-{\bf 1}}\leq 13\cdot 15\d_2=195\d_2,
$$
i.e. $\norm{h^2(t)-{\bf 1} }\leq 195\d_2$.
Then we have $\norm{h(t)-{\bf 1} }\leq 100\d_2$ and finally
$$
\norm{w(t)-\til{w}(t)}=\norm{\til{w}^*(t)w(t)-{\bf 1} }
=\norm{h(t)-{\bf 1} }\leq 100\d_2.
$$
\medskip\noindent
Now we should estimate the commutator $[u\oplus {\bf 1} _{N-n},w(t)]$.
But it is easy to see that
\begin{eqnarray*}
\norm{[u\oplus {\bf 1} _{N-n},w(t)]}&\leq&\norm{[u,w]}\\
&+&\norm{v_{l1}-v_{l1}u_l}+
\norm{u_1v_{l1}-v_{l1}}+\norm{v_{1l}-v_{1l}u_1}+\norm{u_lv_{1l}-v_{1l}}\\
&<&\d_3+2\norm{u_1-{\bf 1} }+2\norm{u_l-{\bf 1} }\leq\d_3+4\sqrt[4]{\e},
\end{eqnarray*}
where $u_1$ and $u_l$ are the first and the last diagonal entries of the
matrix $u$.\q
\begin{lem}\label{2matr.}
Let $w_1,w_2\in U_n({\bf C})$ be almost commuting block 3-diagonal
matrices of the form $($\ref{3diag}$)$, $\norm{[w_1,w_2]}\leq\d_4$.
Define for
both these matrices paths $w_1(t)$ and $w_2(t)$ by $($\ref{bigmatr}$)$.
Then one has
$$
\norm{[w_1(t),w_2(t)]}\leq 13\d_4.
$$
\end{lem}
\noindent
{\bf Proof.}
It can be easily checked that the commutator $[w_1(t),w_2(t)]$ can be
decomposed into sum of the direct sum of smaller commutators $[w_1,w_2]$
and of 12-diagonal matrix with diagonals coinciding with direct sums of
certain diagonals of the matrix $[w_1,w_2]$.\q
\begin{lem}
Let
$$
w'=\left(\begin{array}{cccc}
w_{11}&w_{12}&&w_{1N}\\
w_{21}&w_{22}&\ddots&\\
&\ddots&\ddots&w_{N-1,N}\\
w_{N1}&&w_{N,N-1}&w_{N,N}
\end{array}\right)\in M_N({\bf C})
$$
be a three-diagonal matrix and let
$w'=\til{w}'h'$ be the polar decomposition such that
\be\label{otsen}
\norm{w'-\til{w}'}\leq\d_5.
\ee
Then for any $\d_6>0$ there exist an integer
$r$ $($not depending on $N$$)$ and a decomposition
$$
\til{w}'=w_0+w_1
$$
such that the matrix $w_0$ is $(2r+3)$-diagonal and $\norm{w_1}\leq\d_6$.
\end{lem}
\noindent
{\bf Proof.}
Consider the Taylor formula
\begin{eqnarray*}
\til{w}'&=&w'((w')^*w')^{-1/2}=w'({\bf 1} +((w')^*w'-{\bf 1} ))^{-1/2}\\
&=&w'({\bf 1} +a_1((w')^*w'-{\bf 1} )+a_2((w')^*w'-{\bf 1} )^2+
\ldots+a_r((w')^*w'-{\bf 1} )^r+R_r),
\end{eqnarray*}
where in view of (\ref{otsen}) we have
$$
\norm{R_r}\leq\norm{(w')^*w'-{\bf 1} }^r=\norm{(h')^2-{\bf 1} }^r\leq
(3\d_5)^r.
$$
Take $r$ to satisfy
\be\label{r}
(3\d_5)^r\leq\d_6
\ee
and put
\begin{eqnarray*}
&w_0=w'({\bf 1} +a_1((w')^*w'-{\bf 1} )+a_2((w')^*w'-{\bf 1} )^2+
\ldots+a_r((w')^*w'-{\bf 1} )^r),&\\
&w_1=w'R_r.&
\end{eqnarray*}
Then as $w'$ and $((w')^*w'-{\bf 1} )$ are three-diagonal and
$(2r+1)$-diagonal respectively, so the matrix $w_0$ is
$(2r+3)$-diagonal.\q
\begin{lem}\label{cont}
Let $v'(t)$ be a path in the matrix algebra $M_N({\bf C})$ such that
$$
\dist(v'(t), U_N({\bf C}))\leq\d_7.
$$
Then there exists a unitary path
$v(t)$ such that for every $t$ one has
$$
\norm{v(t)-v'(t)}\leq 3\d_7.
$$
\end{lem}
\noindent
{\bf Proof.}
Let $v'\in M_N({\bf C})$ be a matrix with $\dist(v',U_N({\bf C}))\leq\d_7$,
and let $v'=vh$ be its polar decomposition. By supposition there exists
a unitary $w$ such that $\norm{v'-w}\leq\d_7$. Then $\norm{vh-w}\leq\d_7$,
hence (as in proof of Lemma \ref{bigmatrix}) we get the estimates
\be\label{soprnorm1}
\norm{h-v^{-1}w}\leq\d_7,
\ee
and
\be\label{soprnorm2}
\norm{h-(v^{-1}w)^*}\leq\d_7.
\ee
It follows from (\ref{soprnorm1}) and (\ref{soprnorm2}) that
$$
\norm{v^{-1}w-(v^{-1}w)^*}\leq 2\d_7,
$$
hence
\be\label{soprnorm3}
\norm{v^{-1}w-{\bf 1} }\leq 2\d_7.
\ee
Estimates (\ref{soprnorm1}) and (\ref{soprnorm3}) imply
$\norm{h-{\bf 1} }\leq 3\d_7$,
and finally
$$
\norm{v'-v}=\norm{h-{\bf 1} }\leq 3\d_7.
$$
Now if we have a path $v'(t)$ with the polar decomposition $v'(t)=v(t)h(t)$
then the path $v(t)$ is continuous and $\norm{v(t)-v'(t)}\leq 3\d_7$.\q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Construction of homotopy}
\subsubsection*{Case of two unitaries}\label{paths}
%\setcounter{equation}{0}
In this section we give an explicit construction of the paths $u(t)$ and
$v(t)$ satisfying Theorem \ref{alm->asym}. This construction consists
of four steps.
\medskip\noindent
We start with two unitaries $u,v\in U_n({\bf C})$ with $\norm{[u,v]}\leq\e$
and we assume that $u$ is already diagonal with ordered eigenvalues.
Let $N=nm$ where the number $m$ will be
defined later. The homotopy which we are constructing should begin at the
pair of unitaries $u\oplus {\bf 1} _{N-n}$ and $v\oplus {\bf 1} _{N-n}$.
In fact we will
construct paths $u(t)$ and $v'(t)$ with non-unitary $v'(t)$ but this
latter path will be close to the unitary group,
$\dist(v'(t),U_{N})\leq C\sqrt[4]{\e}$ for some constant $C$.
By Lemma \ref{cont} this would imply
existence of exactly unitary path $v(t)$ with the necessary properties.
At every step of the homotopy we should control two estimates --- the
commutator norm $\norm{[u(t),v'(t)]}$ and the distance from the unitary
group $\dist(v'(t),U_N({\bf C}))$.
\subsubsection*{Step 1}
Put $w=d(v)$ defined by (\ref{tridiag}).
While not changing the first matrix $u\oplus {\bf 1} _{N-n}$, we connect the
matrix $v\oplus {\bf 1} _{N-n}$ by the
linear path with the block 3-diagonal matrix
$w\oplus {\bf 1} _{N-n}$. At this step we have
$$
\norm{[u(t),v'(t)]}\leq 4\sqrt[4]{\e}
\quad{\rm and}\quad
\dist(v'(t),U_N({\bf C}))\leq\norm{v-w}\leq 4\sqrt{\pi}\sqrt[4]{\e}.
$$
\subsubsection*{Step 2}
Connect the unit matrix ${\bf 1}_{N-n}$ with the direct sum
$w\oplus\ldots\oplus
w$ of $m-1$ copies of the matrix $w$. The matrix $u\oplus {\bf 1} _{N-n}$ is
still fixed at the second step of homotopy and both the commutator norm
$\norm{[u(t),v'(t)]}$ and $\dist(v'(t),U_N({\bf C}))$ are
not being changed.
\subsubsection*{Step 3}
We still do not change the matrix $u\oplus {\bf 1} _{N-n}$, and define the
path $v'(t)$ on the third step by $v'(t)=w(t)$, where $w(t)$ is given by
(\ref{bigmatr}). Then by Lemma \ref{bigmatrix} (taking there
$\d_1=4\sqrt{\pi}\sqrt[4]{\e}$ and $\d_2=4\sqrt[4]{\e}$) we get the
estimates
$$
\norm{u\oplus {\bf 1} _{N-n},v'(t)}\leq 8\sqrt[4]{\e}
\quad {\rm and}\quad
\dist(v'(t),U_N({\bf C}))\leq\norm{v'(t)-v(t)}\leq 400\sqrt{\pi}\sqrt[4]{\e}
$$
for $v(t)=v'(t)(v'^*(t)v'(t))^{-1/2}$.
\subsubsection*{Step 4}
At the last step we will not change the second matrix $v(1)$.
The matrix $v'(1)$ is 3-diagonal and
$\norm{v'(t)-v(t)}\leq 400\sqrt{\pi}\sqrt[4]{\e}=\d_5$, so by Lemma
\ref{bigmatrix} for $\d_6=\e/8$ we have a decomposition $v(t)=w_0+w_1$
with $\norm{w_1}\leq\e/8$ and with $w_0$ being $(2r+3)$-diagonal for $r$
satisfying the inequality (\ref{r}), i.e.
$$
(1200\sqrt{\pi}\sqrt[4]{\e})^r\leq\e/8.
$$
The choice of such $r$ is possible
if $\e<\e_0$, where $\e_0$ satisfies $1200\sqrt{\pi}\sqrt[4]{\e_0}\leq 1$.
\medskip\noindent
Then $\norm{[u(t),v(1)]}\leq\norm{[u(t),w_0]}+\e/4$ and it will be
sufficient to find a path $u(t)$ starting from $u\oplus {\bf 1} _{N-n}$ such
that
$$
\norm{[u(1),w_0]}\leq\e/4.
$$
For $t\in [0,1]$ define diagonal paths $u_j(t)\in U_n({\bf C})$ by
$$
u^{(1)}(t)=\left(\begin{array}{ccc}
e^{2\pi i((1-t)\ph_1+t\ph_1/m)}&&\\
&\ddots&\\
&&e^{2\pi i((1-t)\ph_n+t\ph_n/m)}
\end{array}\right)
$$
and
$$
u^{(j)}(t)=\left(\begin{array}{ccc}
e^{2\pi it\ph_1j/m}&&\\
&\ddots&\\
&&e^{2\pi it\ph_nj/m}
\end{array}\right)
,\quad j=2,\ldots,m.
$$
Put $u(t)=u^{(1)}(t)\oplus\ldots\oplus u^{(m)}(t)\in U_N({\bf C})$.
Then the diagonal matrix $u(1)$ will consist of all ordered power $m$
roots of eigenvalues $e^{2\pi i\ph_i}$ of the matrix $u$.
\medskip\noindent
Denote by $d_k(w_0)$ the diagonal of the matrix $w_0$ lying $k$ lines above
(or below if $k$ is negative) the main diagonal $d_0(w_0)$.
Then
$$
w_0=\sum_{k=-r-1}^{r+1}d_k(w_0)
$$
and $\norm{d_k(w_0)}\leq 1+\e/8$.
It is easy to see (cf. remark \ref{lacunas} in the case of lacunas in
the spectrum of $u$) that
$$
\norm{[u(t),d_k(w_0)]}\leq (1+\e/8)\cdot 2k\sqrt[4]{\e}
\cdot\left(1-t+\frac{t}{m}\right),
$$
therefore
$$
\norm{[u(t),w_0]}\leq
2\,(1+\e/8)\left(\sum_{k=1}^{r+1}k\right)\sqrt[4]{\e}\
\left(1-t+\frac{t}{m}\right)
\leq 2\,(1+\e/8)\ (r+1)^2\ \sqrt[4]{\e}\ \left(1-t+\frac{t}{m}\right).
$$
So along the whole path we have the necessary estimate
$$
\norm{[u(t),v(t)]}\leq 3(r+1)^2\sqrt[4]{\e}.
$$
To finish the construction of homotopy we have to choose the number $m$ so
that it would satisfy the inequality
$$
2\,(1+\e/8)\ (r+1)^2\ \sqrt[4]{\e}\ \frac{1}{m}<\e/4,
$$
then we will have $\norm{[u(1),w_0]}\leq\e/4$, hence
$\norm{[u(1),v(1)]}\leq\e/2$. The case of two almost commuting unitaries is
proved.
\subsubsection*{Case of free abelian group}
Now we can prove the general case of $k$ almost commuting unitaries
$u_1,\ldots,u_k$, $\norm{[u_i,u_j]}\leq\e$. As in the two matrices case
we diagonalize the first matrix $u_1$ and repeat the previous construction
for the other unitaries $u_2,\ldots,u_k$
(using Lemma \ref{2matr.} we can control the commutator norm for these
unitaries). Then we will get the unitary
paths $u_i(t)$ such that $u_i(0)=u_i\oplus {\bf 1} _{N-n}$,
$u_i(1)=u_i^{(1)}$,
with properties
\begin{enumerate}
\item
$\norm{[u_1^{(1)},u_j^{(1)}]}\leq\e'$,\quad $j\neq 1$,
\vspace{-\itemsep}
\item
$\norm{[u_i^{(1)},u_j^{(1)}]}\leq C\sqrt[4]{\e}$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}\leq C\sqrt[4]{\e}$
\end{enumerate}
with small enough $\e'>0$ which we define below and with some constant
$C$. Then at the second step we diagonalize the matrix $u_2^{(1)}$ and
repeat the above construction to the matrices
$u_1^{(1)},u_3^{(1)},\ldots,u_k^{(1)}$. Then we will get the matrices
$u_1^{(2)},\ldots,u_k^{(2)}$ satisfying
\begin{enumerate}
\item
$\norm{[u_2^{(2)},u_j^{(1)}]}\leq\e'$,\quad $j\neq 2$,
\vspace{-\itemsep}
\item
$\norm{[u_1^{(2)},u_j^{(2)}]}\leq C\sqrt[4]{\e'}$,\quad $j\neq 1,2$,
\vspace{-\itemsep}
\item
$\norm{[u_i^{(2)},u_j^{(2)}]}\leq C^2\sqrt[16]{\e}$,\quad $i,j\neq 1,2$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}\leq C^2\sqrt[16]{\e}$.
\end{enumerate}
Repeating this procedure $k-1$ times we finally get the matrices
$u_1^{(k-1)},\ldots,u_k^{(k-1)}$ with properties
\begin{enumerate}
\item
$\norm{[u_1^{(k-1)},u_j^{(k-1)}]}\leq C^{k-2}(\e')^{(1/4)^{k-2}}$,
\quad $j>1$,
\vspace{-\itemsep}
\item
$\norm{[u_2^{(k-1)},u_j^{(k-1)}]}\leq C^{k-3}(\e')^{(1/4)^{k-3}}$,
\quad $j>2$,\quad etc.,
\vspace{-\itemsep}
\item
$\norm{[u_{k-1}^{(k-1)},u_k^{(k-1)}]}\leq\e'$,
\vspace{-\itemsep}
\item
$\norm{[u_i(t),u_j(t)]}\leq C^{k-1}\e^{(1/4)^{k-1}}$.
\end{enumerate}
So, we finally have $\norm{[u_i^{(k-1)},u_j^{(k-1)}]}\leq
C^{k-2}(\e')^{(1/4)^{k-2}}$ for all $i,j$.
Taking $\e'$ to satisfy the inequality
$C^{k-2}(\e')^{(1/4)^{k-2}}\leq\e/2$ we obtain the
statement of the theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{Case of a general abelian group}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
In this section we have to deal with the torsion generators
$v_1,\ldots,v_l$. In the following
we refer to the paper \cite{Grov} though it is possible to give the
explicit construction.
\begin{thm}%[\cite{Grov}]
For any $\e$ and for any set of unitaries $v_1,\ldots,v_l\in U_n({\bf C})$
with $\norm{[v_i,v_j]}\leq\e$, $\norm{v_j^{n_j}-{\bf 1} }\leq\e$ there exists
$\d(\e)$ and unitaries $v'_1,\ldots,v'_l\in U_n({\bf C})$ such that
$\d(\e)$ tends to zero when $\e\to 0$ and $[v'_i,v'_j]=0$,
$(v'_j)^{n_j}={\bf 1} $. \q
\end{thm}
\noindent
If $u_1,\ldots,u_k,v_1,\ldots,v_l$ is an $\e$-almost representation
of the abelian group $\G$
then we can connect the unitaries $v_1,\ldots,v_l$ with
$v'_1,\ldots,v'_l$ and afterwards we will have
$\norm{[u_i,v'_j]}\leq\e+2\d=\d'$.
Denote by $p_r$ the spectral projections
for the set of commuting unitaries $v'_j$ and let $N$ be the minimal
common divisor of the degrees $n_j$, $j=1,\ldots,l$. We can assume that
$\e$ is much smaller than the minimal distance between eigenvalues of the
unitaries $v'_1,\ldots,v'_l$ which equals to $|e^{2\pi i/N}-1|$. Let
$u_{i;rs}=p_ru_ip_q$ be the matrix blocks of $u_i$ with respect to the
decomposition $\oplus_r p_r={\bf 1} $. Then it is easy to see from the
estimate $\norm{[u_i,v'_j]}\leq\d'$ that
\be\label{grub}
\norm{u_{i;rs}}\leq\frac{\d'}{|e^{2\pi i/N}-1|},\quad r\neq s.
\ee
Denote now by $\til{u}_i$ the diagonal matrices consisting of the diagonal
entries of $u_i$, $\til{u}_i=\diag\{u_{i;11},u_{i;22},\ldots\}$. Then
by summing the inequalities (\ref{grub})
one gets the following (very rough) estimate:
\begin{lem}
One has $\norm{u_i-\til{u}_i}\leq N^2\frac{\d'}{|e^{2\pi i/N}-1|}$. \q
\end{lem}
\noindent
So the distance from $\til{u}_i$ to the unitary group is small enough, hence
there exist unitaries $u'_{i;rr}\in p_r M_n p_r$ with
$\norm{u_{i;rr}-u'_{i;rr}}\leq N^3\d'=\d''$
(remember that $N$ does not depend on dimension of the almost
represenation). Put $u'_i=\diag\{u'_{i;11},u'_{i;22},\ldots\}$. Then
$\norm{u'_i-u_i}\leq \d''$, so we have $\norm{[u'_i,u'_j]}\leq 2\d''$, and
as the matrices $u'_i$ are block-diagonal, so they exactly commute with
$v'_j$: $[u'_i,v'_j]=0$.
\medskip\noindent
Now we have a $2\d''$-almost representation of the group $\G$ where all
relations hold {\it exactly} except the commutators $[u'_i,u'_j]$.
After connecting $u_i$ with $u'_i$ and $v_j$ with $v'_j$ we can
proceed as in the case of free abelian groups. Let take a basis such that
the matrix $u'_1$ is diagonal with eigenvalues being ordered.
Then as $v'_j$ exactly
commute with $u'_i$, so they are block-diagonal,
$$
v'_j=\diag\{v'_{j;11},v'_{j;22},\ldots\}
$$
and every diagonal entry satisfies $(v'_{j,rr})^{n_j}={\bf 1} $.
After applying our construction of diminishing commutators by
increasing the dimension
we can for any $\e''>0$ get matrices
$u''_i,v''_j$ of bigger dimension with the properties
$$
[u''_i,v''_j]=[v''_i,v''_j]=(v''_j)^{n_j}-{\bf 1} =0,
$$
$$
\norm{[u''_1,u''_i]}\leq\e'',\quad {\rm for}\ \ i>1
$$
and
$$
\norm{[u''_i,u''_j]}\leq C\sqrt[4]{\d''}.
$$
Then proceeding in the same way with $u_2,u_3,$ etc. we get the necessary
paths. \q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Case of fundamental groups of oriented surfaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\G=\la a_1,b_1,\ldots,a_m,b_m|a_1b_1a_1^{-1}b_1^{-1}\cdot\ldots\cdot
a_mb_ma_m^{-1}b_m^{-1}\ra$. Let $\s$ be an $\e$-almost representation of
$\G$, $u_i=\s(a_i)$, $v_i=\s(b_i)$, $u_i,v_i\in U_n$, $i=1,\ldots,m$.
Denote $\g(u,v)=uvu^{-1}v^{-1}$, then we have
$$
\norm{\g(u_1,v_1)\cdot\ldots\cdot \g(u_m,v_m)-I}\leq\e.
$$
Consider the map
\begin{equation}\label{map_gamma}
\g:U_n\times U_n\arr SU_n.
\end{equation}
To prove the asymptotic stability property for fundamental groups of
oriented two-dimensional manifolds we have to use the following elementary
statement about the map (\ref{map_gamma}).
\begin{lem}\label{lem:podniat}
Let $(u_0,v_0)\in U_n\times U_n$ and let $c(t)\in SU_n$, $t\in [0,1]$, be
a path such that $\g(u_0,v_0)=c(0)$. Then for any $\d>0$ there
exists a path $(u_t,v_t)\in U_n\times U_n$ such that
$\norm{\g(u_t,v_t)-c(t)}<\d$.
\end{lem}
{\bf Proof.}
Remember that a pair $(u,v)\in U_n\times U_n$ is called {\it irreducible}
if there is no common invariant subspace for $u$ and $v$.
It was shown in \cite{exel} that the set of regular points for the map
$\g$ (\ref{map_gamma}) coincides with the set of irreducible pairs.
Denote the set of reducible pairs $(u,v)\in U_n\times U_n$ by $S$.
For any $k=1,\ldots,n-1$ by $\Sigma_k\subset SU_n$ denote the set of
block-diagonal matrices
$\left(\begin{array}{cc}c_1&0\\0&c_2\end{array}\right)$ with respect to
some invariant subspace $V$, $\dim V=k$, such that $c_1\in SU_k$, $c_2\in
SU_{n-k}$. Put $\Sigma=\cup_k\Sigma_k\subset SU_n$. Then obviously
$\g(S)\subset \Sigma$. Notice that every $\Sigma_k$ is a submanifold
in $SU_n$ with codimension one. So $\Sigma$ divides $SU_n$ into a finite
set of closed path components $M_j$, $\cup_j M_j=SU_n$, and for every point
$c\in M_j$ the set $\g^{-1}(c)$ consists only of regular points. Hence
every path in $M_j$ transversal to its boundary can be lifted up to a path
in $U_n\times U_n$ with a fixed starting point.
Without loss of generality we can assume that the path $c(t)$ is
transversal to every $\Sigma_k$. Let $t_0\in \{c(t)\}\cap\Sigma_k$. It
remains to show that we can lift the path $c(t)$ in some neighborhood of
the point $c_0=c(t_0)$. Let $(u_0,v_0)\in U_n\times U_n$ be such point
that $\g(u_0,v_0)=c_0$. If the point $(u_0,v_0)$ is a regular point then
the statement is obvious. Otherwise we can write
$$
u_0=\left(\begin{array}{cc}u_1&0\\0&u_2\end{array}\right),\qquad
v_0=\left(\begin{array}{cc}v_1&0\\0&v_2\end{array}\right)
$$
with respect to some basis and we can assume that matrices $v_1$ and $v_2$
are diagonal. Let
$$
e^{2\pi i\ph_1},\ldots,e^{2\pi i\ph_k}\quad{\rm and}\quad
e^{2\pi i\ph_{k+1}},\ldots,e^{2\pi i\ph_n}
$$
be the eigenvalues of
$v_1$ and $v_2$ respectively.
Slightly changing $v_0$ we can assume that for all $i=2,\ldots,k$,
$j=k+2,\ldots,n$ the values $\ph_i-\ph_1$, $\ph_j-\ph_{k+1}$ differ from
each other. Multiplying $v_1$ by $e^{-2\pi i\ph_1 t}$ and $v_2$ by
$e^{-2\pi i\ph_{k+1} t}$, $t\in [0,1]$, we connect the matrix $v_0$ with
the matrix
$$
v'_0=\left(\begin{array}{cc}v'_1&0\\0&v'_2\end{array}\right)=
\left(\begin{array}{cc}e^{-2\pi i\ph_1}v_1&0\\
0&e^{-2\pi i\ph_{k+1}}v_2\end{array}\right)
$$
which has two eigenvalues equal to one and all other eigenvalues being
different from each other. Obviously the value
$\g(u_0,v_0)=\g(u_0,v'_0)=c_0$ does not change along this path.
Denote by $e_1,\ldots,e_n$ the basis consisting of the eigenvalues of
$v'_0$ and let $w(t)\in U_n$, $t\in[0,1]$, be a rotation of the vectors
$e_1$ and $e_{k+1}$:
$$
w(t)e_1=\cos t e_1-\sin t e_{k+1},\quad
w(t)e_{k+1}=\sin t e_1+\cos t e_{k+1},\quad
w(t)e_j=e_j \quad {\rm for}\quad j\neq 1,k+1.
$$
Obviously $w(t)$ commutes with $v'_0$. Put $u_t=u_0w(t)$. Then
$$
\g(u_t,v'_0)=u_0w(t)v'_0w^{-1}(t)u_0^{-1}(v'_0)^{-1}=\g(u_0,v'_0)=c_0
$$
and for $\sin t\neq 0$ the pair $(u_t,v'_0)$ is {\it irreducible}
(since $v'_0$ is diagonal with
only two coinciding eigenvalues, so its invariant subspaces are easy to
describe, then it is easy to check that they are not invariant under the
action of $u_t$) with the same value of $\g$. Then it is possible to
extend the path $(u_t,v_t)$ through the point $c_0$. \q
\begin{prop}\label{homot_to_1}
Any $\e$-almost representation of $\G$ is homotopically equivalent in
${\cal R}_{2\e}(\G)$ to an $2\e$-almost representation with
$u_2=v_2=\ldots=u_m=v_m=I$.
\end{prop}
{\bf Proof.}
Connect the matrix $\g(u_m,v_m)$ with $I$ by a path $c_m(t)$. Then by the
Lemma \ref{lem:podniat} we can find a path $(u_m(t),v_m(t))\in U_n\times
U_n$ such that
$$
\norm{\g(u_m(t),v_m(t))-c_m(t)}\leq\frac{\e}{2m}.
$$
Notice
that the set $\g^{-1}(I)=\{(u,v):uv=vu\}$ is path-connected, so we can
assume that the end point of the path $(u_m(t),v_m(t))$ is $(I,I)$.
Put
$$
c_{m-1}(t)=\g(u_{m-1},v_{m-1})c^{-1}_m(t).
$$
Again by the Lemma
\ref{lem:podniat} we can find a path $(u_{m-1}(t),v_{m-1}(t))\in
U_n\times U_n$ such that
$$
\norm{\g(u_{m-1}(t),v_{m-1}(t))-c_{m-1}(t)}\leq\frac{\e}{2m}.
$$
Then
$$
\norm{\g(u_1,v_1)\cdot\ldots\cdot\g(u_{m-1}(t),v_{m-1}(t))
\cdot\g(u_m(t),v_m(t))-I}\leq\e+\frac{\e}{m}
$$
and at the end point we have $(u_m(t),v_m(t))=(I,I)$. Proceeding by
induction we finish the proof.\q
It now follows from the proposition \ref{homot_to_1} that the
asymptotic stability property
for the group $\G$ follows from the same property for the group ${\bf Z}^2$
with generators $u_1,v_1$. \q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example of a group without asymptotic stability property}
\label{example_of_a_group}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
To show that not every group is asymptotically stable we give the
following example.
Consider the group $\G=\la a,b,c|aca^{-1}c^{-1},b^2,(ab)^2\ra$.
\begin{thm}\label{primergruppy}
The group $\G$ is not asymptotically stable.
\end{thm}
{\bf Proof.}
Put $\o=e^{2\pi i/n}$ and define a
family of almost representations $\s_n$ taking values in $U_n$ by
$$
\s_n(a){=}\!\left(\begin{array}{cccccc}
\o&&&&&\\
&\o^2&&&&\\
&&\cdot&&&\\
&&&\cdot&&\\
&&&&\cdot&\\
&&&&&\o^n
\end{array}\right)\!\!,\
%\quad
\s_n(c){=}\!\left(\begin{array}{cccccc}
0&&&&&1\\
1&0&&&&\\
&1&\cdot&&&\\
&&\cdot&\cdot&&\\
&&&\cdot&\cdot&\\
&&&&1&0
\end{array}\right)\!\!,\
%\quad
\s_n(b){=}\!\left(\begin{array}{cccccc}
&&&&&1\\
&&&&1&\\
&&&\cdot&&\\
&&\cdot&&&\\
&\cdot&&&&\\
1&&&&&
\end{array}\right)\!\!.
$$
Here the matrices $\s_n(a)$ and $\s_n(c)$ are the Voiculescu matrices
\cite{voi} with the winding number \cite{e-l} equal to one, and one has
$$
\s_n(a)\s_n(c)\s_n(a)^{-1}\s_n(c)^{-1}=\o\cdot I,\quad
\s_n(b)^2=I,\quad (\s_n(a)\s_n(b))^2=\o\cdot I,
$$
so
$$
\e_n=\nnn{\s_n}=|\o-1|\to 0 \quad{\rm when}\ \ n\to\i,
$$
hence for every $\e>0$ there exists an $\e$-almost representation $\s$ of
$\G$ such that the winding number of the pair $(\s(a),\s(c))$ equals one.
\smallskip
Suppose the opposite, i.e. that the group $\G$ is asymptotically stable.
Then there should be such $\e$ that the number $\d(\e)\leq 1$. Take such
$\e$ and an $\e$-almost representation $\s_0$ with a non-zero winding
number of the pair $(\s_0(a),\s_0(c))$.
\smallskip
By supposition there exists an asymptotic representation
$\s_t\in{\cal R}_{asym}(\G)$ extending $\s_0$ such that
$\nnn{\s_t}\leq 1$ for all $t\in [0,\i)$ and for any $\e'>0$ there exists
$t_0$ such that $\nnn{\s_{t_0}}\leq \e'$. Fix this $t_0$ and denote
$\s_{t_0}$ by $\s$. Let $n$ and $n+m$ be the dimension of the almost
representation $\s_0$ and $\s$ respectively.
Then one has
$$
\norm{\s(a)\s(c)-\s(c)\s(a)}\leq\e',
\quad
\norm{\s(b)^2-I}\leq\e',
\quad
\norm{\s(a)\s(b)\s(a)\s(b)-I}\leq\e'.
$$
Notice that as along the whole path $\s_t$ one has
$$
\norm{\s_t(b)^2-I}\leq\nnn{\s_t}\leq\d(\e)\leq 1,
$$
so the eigenvalues of $\s_t(b)$ satisfy the estimate $|\l^2-1|\leq 1$,
hence the number of eigenvalues $\l\in\Sp \s_t(b)$ with $\Re\l<0$
does not change along the whole path $\s_t$ in $U_\i$,
therefore the number of eigenvalues $\l\in\Sp\s(b)$ with $|\l+1|\leq \e'$
cannot exceed $n$ (the maximal number of eigenvalues with $\Re\l<0$ of
$\s_0(b)$) and the number of eigenvalues with $|\l-1|\leq \e'$
is not less than $m$.
Then there exists a matrix $\s(b)'\in U_{n+m}$ such that
$\norm{\s(b)-\s(b)'}\leq\e'$ and that the matrix $\s(b)'$ has not more
than $n$ eigenvalues equal to $-1$ and not less than $m$ eigenvalues
equal to $1$. Then we have
\begin{equation}\label{1*}
\norm{\s(a)\s(b)'\s(a)\s(b)'-I}\leq 3\e'.
\end{equation}
Notice that $(\s(b)')^2=I$ and
\begin{equation}\label{1a*}
|\tr(\s(b)')|\geq m-n.
\end{equation}
Let
$$
\s(a)=\left(\begin{array}{ccc}
\o_1&&\\
&\ddots&\\
&&\o_{n+m}
\end{array}\right)
$$
be the matrix of the operator $\s(a)$ in the basis consisting of its
eigenvectors. It was shown in \cite{manFA} that if the winding
number of the pair $(\s(a),\s(c))$ is non-zero then for any $\e'>0$ there
exists $\d'(\s')$ such that $\d'(\e')\to 0$ when $\e'\to 0$ and that all
lacunae in $\Sp\s(a)$ do not exceed $\d'(\e')$.
Denote the number of eigenvalues of $\s(a)$ with $|\Im\o_j|>2\e'$ by $N$.
Then we have
\begin{equation}\label{1d*}
N\geq\frac{2\pi-10\e'}{\d'(\e')}.
\end{equation}
As $(\s(b)')^2=I$, so it follows from (\ref{1*}) that
\begin{equation}\label{1c*}
\norm{\s(a)\s(b)'-\s(b)'\s(a)^*}\leq 3\e'.
\end{equation}
Denote by $b_{ij}$ the matrix elements of the matrix $\s(b)'$.
It follows from (\ref{1c*}) that all matrix elements of
$\s(a)\s(b)'-\s(b)'\s(a)^*$ do not exceed $3\e'$, i.e.
\begin{equation}\label{2*}
|b_{ii}(\o_i-\ov{\o}_i)|\leq 3\e', \quad i=1,\ldots,n+m.
\end{equation}
Let us estimate $\tr(\s(b)')$.
We have
$$
|\tr(\s(b)')|=\left|\sum_{i=1}^{n+m}b_{ii}\right|\leq
\sum_{i=1}^{n+m}|b_{ii}|=\sum\nolimits'|b_{ii}+\sum\nolimits''|b_{ii}|,
$$
where $\sum'$ denotes the sum for those numbers $i$ for which one has
$|\Im\o_i|>2\e'$ and $\sum''$ is the sum for the remaining numbers.
As for all $i$ one has $|b_{ii}|\leq 1$, so the last sum do not exceed
the number of summands,
$$
\sum\nolimits''|b_{ii}|\leq n+m-N.
$$
It follows from (\ref{2*}) that for those $i$ which are included into the
first sum we have $|\o_i-\ov{\o}_i|>4\e'$, hence those $b_{ii}$ satisfy
$$
|b_{ii}|<\frac{3}{4},
$$
so
$$
\sum\nolimits'|b_{ii}|<\frac{3}{4}N,
$$
hence we have
$$
|\tr(\s(b)')|<\frac{3}{4}N+n+m-N=n+m-\frac{N}{4}
$$
and it follows from (\ref{1d*}) that
\begin{equation}\label{2a*}
|\tr(\s(b)')|2n}$, then (\ref{1d*}) and
(\ref{2a*}) give a contradiction. \q
\begin{cor}\label{RG=RH}
Let $\s_t\in{\cal R}_{asym}(\G)$ be an asymptotic representation. Then the
winding number of the pair $(\s_t(a),\s_t(c))$ is zero for big enough $t$.
In particular, it means that $\s_t$ is homotopic to an asymptotic
representation $\rho_t\in{\cal R}_{asym}(\G)$ with $\rho_t(c)=I$ in the
class of asymptotic representations. \q
\end{cor}
Denote by $H$
the subgroup $\la a,b|b^2,(ab)^2\ra\cong {\bf Z}_2\ast{\bf Z}_2\subset\G$.
Then $\G\cong {\bf Z}^2\ast_{\bf Z}H$, so we have $B\G={\bf T}^2\cup BH$,
$S^1={\bf T}^2\cap BH$, where $B\G$, $BH$, $S^1$ and ${\bf T}^2$ are the
classifying spaces of the groups $\G$, $H$, ${\bf Z}$ and ${\bf Z}^2$
respectively, and the inclusion $S^1\subset {\bf T}^2$ is the standard
inclusion onto the first coordinate. Then one has an exact sequence
\begin{equation}\label{K6}
\diagram
K^0(B\G)\rto & K^0(BH)\oplus K^0({\bf T}^2)\rto & K^0(S^1)\dto\\
K^1(S^1)\uto & K^1(BH)\oplus K^1({\bf T}^2)\lto & K^1(B\G)\lto
\enddiagram
\end{equation}
and as the maps $K^*({\bf T}^2)\arr K^*(S^1)$ are onto, so the vertical
maps in (\ref{K6}) are zero and the group $K^0(B\G)$ contains an element
\begin{equation}\label{beta}
\beta\in K^0(B\G),
\end{equation}
which is mapped onto the Bott generator of $K^0({\bf T}^2)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Almost representations of groups $\pi\times{\bf Z}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
Let $\pi$ be a discrete finitely presented group and
fix a number $\e>0$. Let also $F\subset\pi$ be a
finite subset. 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$.
But in this section we are interested in
a special class of almost representations of the groups of the form
$\pi\times\bf Z$ such that they are genuine representations when
restricted to the group $\pi$ \cite{M-stekl99}.
We will see that these almost representations behave in some sense 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}
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.
$$
}
\end{dfn}
Let $X$ be a compact metric space with metric $d$.
\begin{dfn}
{\rm
An action $\a$ of $\pi$ on $X$ is called an {\em $\e$-dense finite action}
(with respect to $F$) if the following properties hold:
\begin{enumerate}
\vspace{-\itemsep}
\item
for any $g\in F$ one has $d(\a_g(x),x)<\e$ for all $x\in X$;
\vspace{-\itemsep}
\item
there exists a point $x_0\in X$ such that its orbit $\{\a_g(x_0):g\in\pi\}$
is finite;
\vspace{-\itemsep}
\item
the orbit $\{\a_g(x_0):g\in\pi\}$ is an $\e$-net in $X$, i.e. for any
$x\in X$ one can find $g\in\pi$ such that $d(x,\a_g(x_0))<\e$.
\vspace{-\itemsep}
\end{enumerate}
}
\end{dfn}
We now are going to give a construction of almost representations of
the group $\pi\times{\bf Z}$ (with respect to $F$).
\medskip\noindent
Let $U_q(C(X))$ be the the set of $U_q({\bf C})$-valued
continuous functions on $X$ and let
$u\in U_q(C(X))$ be a unitary element. Then there exists a constant
$C$ such that one has
\be\label{boundedness}
\norm{u(x)-u(y)}0$.
\begin{rmk}
{\rm
In the case when $q=1$ (i.e. when the element $u$ defines a vector bundle
over $X\times S^1$ of rank one) every $\e$-almost representation of
$\pi\times{\bf Z}$ with small enough $\e$ generates an asymptotic
representation of this group. This statement can be proven by the method
of \cite{M-izv99}. Unfortunately this method cannot be applied to get the
same result for arbitrary $q$ and it is unknown if {\it any} almost
representation of $\pi\times{\bf Z}$ generates its asymptotic
representation.
}
\end{rmk}
We give a simple construction of almost
representations of $\pi\times{\bf Z}$ out of representations of $\pi$.
Let $\rho_t$, $t\in S^1$, be a continuous
loop of finitedimensional representations
of $\pi$. Then an almost representation $\sigma$ of
$\pi\times{\bf Z}$ can be
defined by the matrices
$$
\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),
$$
where $g\in\pi$ and $\omega=e^{2\pi i/n}\in S^1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Construction of bundles over $B\pi\times S^1$}\label{bundles}
\label{sect2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have now to repeat the construction of vector bundles over
classifying spaces out of almost representations from the section
\ref{construction} for the case of special almost representations of
groups $\pi\times{\bf Z}$.
Let $E\pi\times{\bf R}$ be the total space of the universal
$\pi\times{\bf Z}$-bundle over $B\pi\times S^1$ and let $y\in E\pi$,
$s\in{\bf R}$. For simplicity sake we assume in this section that $B\pi$
is finite and put $F=\{g\in\pi:g(\bar X)\cap\bar X\neq\emptyset\}$, where
$\bar X\subset E\pi$ is a fundamental domain.
Consider a trivial vector bundle $E\pi\times{\bf
R}\times H_n$ over $E\pi\times{\bf R}$. Starting from a $\d$-almost
representation $\sigma_n$ of the group $\pi\times{\bf Z}$ (with small
enough $\d$) we are going to
construct for $g\in\pi$, $b\in{\bf Z}$ transition functions
$$
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
$$
i.e. $T_{(g,a)}(y)$ are close to the unitary group when $\d$ is small
enough.
Henceforth we can extend the functions $T_{(g,a)}(y)$ to the inner points
of the simplex $\Delta$ by linearity. It is easy to check that the
conditions (\ref {composition}) are satisfied. So we have constructed a
vector bundle over $B\pi\times S^1$. This bundle is a representative of an
element of the group $K^0(B\pi\times S^1)\cong K^0(B\pi)\oplus K^1(B\pi)$.
It is easily seen that the first summand is given by the transition
functions (\ref{triv}) and is a locally flat bundle. Further on we will be
more interested in the second summand.
Remark that the above construction gives for every $\e$-dense finite
action of a group $\pi$ on $X$ a homomorphism $K^1(X)\arr K^1(B\pi)$ which
maps $U_q({\bf C})$-valued functions (= maps from $X$ into $U_q({\bf C})$
= elements of $K^1(X)$) into the constructed vector bundles over
$B\pi\times S^1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Almost representations and subgroups of finite index}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\pi'\subset\pi$ be a subgroup of finite index $m=\#(\pi/\pi')$.
Suppose that the universal covering $E\pi$ of the classifying space $B\pi$
is a finitedimensional simplicial complex with a $\pi$-invariant metric
$d$. Denote by $\ov{Y}_\pi$ (resp. $\ov{Y}_{\pi'}$) the fundamental
domain for the action of the group $\pi$ (resp. $\pi'$) on $E\pi$,
$\ov{Y}_\pi\subset\ov{Y}_{\pi'}\subset E\pi$.
Let $p_\pi$ (resp. $p_{\pi'}$) denote the natural projection of the
fundamental domain onto the classifying space of the group $\pi$ (resp.
$\pi'$). We suppose that the metric $d$ is normed by
${\rm diam}(\ov{Y}_\pi)=1$.
The metric $d$ induces in a natural way the metrics on the classifying
spaces $B\pi$ and $B\pi'$. Suppose that we have a map $\ov{f}:
\ov{Y}_{\pi'}\arr\ov{Y}_\pi$ such that it induces a map $f:B\pi'\arr B\pi$
so that we have a commutative diagram
$$
\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}
Note that for 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}
{\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
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Asymptotic and Fredholm representations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
In this section we construct a $C^*$-algebra which is a target for the
theory of asymptotic representations and its embedding into the Calkin
algebra which induces an isomorphism of the $K_1$-groups. As a
consequence we show that every vector bundle over a classifying space
$B\pi$ which can be obtained from an asymptotic representation of a
discrete group $\pi$ can be obtained also from a representation of the
group $\pi\times{\bf Z}$ into the Calkin algebra. We give also a
generalization of the notion of Fredholm representation and show that
asymptotic representations can be viewed as asymptotic Fredholm
representations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic $C^*$-algebra}\label{subsec1.2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is well known that the asymptotic representations are exact
representations in some more compound $C^*$-algebras
~\cite{c-hig}.
Remind this construction. Let
$M_{n}=M_n({\bf C})$
be the
$n\times n$
complex matrix algebra.
Fix a strictly increasing sequence
$n_k$.
Assume that the matrix algebra
$M_{n_{k}}$
is included in natural way into
$M_{n_{k+1}}$:
\begin{displaymath}\label{eq(1)8}
M_{n_{k}}\ni v \mapsto v\oplus\b{0}_{n_{k+1}-n_{k}}\in
M_{n_{k}}\oplus M_{n_{k+1}-n_{k}}\subset M_{n_{k+1}}.
\end{displaymath}
Denote by
$M_{f}$
the sequnence of matrix algebras
$\{M_{n_{k}}\}$.
Consider the
$C^*$-algebra
\begin{displaymath}\label{eq(1)9}
B=B(M_{f}({\bf C}))=\prod_{k=1}^\infty M_{n_k}({\bf C})
\end{displaymath}
which consists of norm bounded sequences of matrices. Denote by
$B^+$
the
$C^*$-algebra
$B$
with adjoint unit.
Both algebras
$B$
and
$B^+$
contain the
$C^*$-ideal
\begin{displaymath}\label{eq(1)10}
I=\bigoplus^{\infty}_{k=1} M_{n_k}({\bf C})
\end{displaymath}
consisting of sequences of matrices, which norm tends to zero.
\medskip
Denote the quotient algebras by
$Q=B/I$
and
$Q^+=B^+/I$,
respectively and denote the corresponding projections by
\begin{displaymath}\label{eq(1)11}
q:B\mapr{}Q ,\qquad
q^{+}:B^{+}\mapr{}Q^{+}
\end{displaymath}
Let
\begin{displaymath}\label{eq(1)13}
\bar\alpha:B^+\arr B^+
\end{displaymath}
be the right shift
\begin{eqnarray*}\label{eq(1)14}
\bar\alpha(m_1,m_2,\ldots)&=&(0,m_1,m_2,\ldots),\quad (m_i)\in B\subset B^+,
\nonumber \\
\bar\alpha ({\bf 1}) &=& {\bf 1}.
\end{eqnarray*}
As
$\bar\alpha(I)\subset I$,
the homomorphism
$\bar\alpha$
induces the homomorphism
$\alpha:Q^+\arr Q^+$.
Let
\begin{displaymath}\label{eq(1)15}
Q^+_\alpha=Q^+_\alpha(M_{f}({\bf C}))=\{q\in
Q^+: \alpha(q)=q\}\subset Q^+
\end{displaymath}
be
$\alpha$-invariant
$C^*$-subalgebra.
The algebra
\begin{displaymath}\label{eq(1)16}
Q_\alpha=Q_{\alpha }(M_{f}({\bf C}))=\{q\in
Q: \alpha(q)=q\}\subset Q
\end{displaymath}
is defined in similar way.
Both algebras
$Q_{\alpha }$
and
$Q^+_{\alpha }$
are quotient algebras of their inverse images
\begin{eqnarray*}\label{eq(1)17}
B_{\alpha }&=&q^{-1}\ll(Q_{\alpha }\rr),
\nonumber\\
B^+_{\alpha }&=&(q^{+})^{-1}\ll(Q^+_{\alpha }\rr),
\end{eqnarray*}
the algebra
$B_{\alpha}$
consists of such sequences
\begin{displaymath}\label{eq(1)18}
(m_1,m_2,\ldots,m_{i},\ldots)\in B\subset B^+,
\end{displaymath}
that
\begin{displaymath}\label{eq(1)19}
\lim\|m_{i+1}-m_{i}\|=0.
\end{displaymath}
\medskip
We will write elements of the algebra $Q^+_\alpha$ as pairs
$(\lambda,(m_i))$, where $\lambda\in{\bf C}$, $(m_i)\in Q_\alpha$.
Adjoined unit gives a split short exact sequence
\begin{equation}\label{eq(1)20}
Q_\alpha\arr Q^+_\alpha\mapr{\epsilon}{\bf C},
\end{equation}
where
$\epsilon$ is the augmentation which
an element of the algebra
$Q^{+}_{\alpha}$
associates with its first component.
%????? (discrete version of asymptotic representations)
Then an asymptotic representation
$\sigma$
defines the homomorphism
\begin{equation}\label{eq(1)21}
\ov{\sigma}:\pi\mapr{}Q^{+}_{\alpha}
\end{equation}
by the formula
\begin{equation}\label{eq(1)22}
\ov{\sigma}(g)=q^{+}\ll(1,\ll\{\sigma_{k}(g)-\b{E}_{n_{k}}\rr\}\rr).
\end{equation}
Notice that as the group $\pi$ is finitely presented, so it follows
from the definition of asymptotic representations that
$\ov{\sigma }(g)\in Q^{+}_{\alpha }$
for arbitrary element
$g\in\pi $.
It is clear that the composition
\begin{displaymath}\label{eq(1)23}
\epsilon\circ\ov{\sigma }:\pi\mapr{}Q^{+}_{\alpha}\mapr{}{\bf C}
\end{displaymath}
maps the whole group
$\pi $
into the unit.
The homomorphism
(\ref{eq(1)21})
is extended in natural way to a unital symmetric homomorphism
(or $*$-homomorphism) of the algebras
\begin{equation}\label{eq(1)24}
\ov{\sigma}:C^*[\pi]\arr Q^+_\alpha.
\end{equation}
Inversely let us consider an unital symmetric homomorphism
(\ref{eq(1)24}), whose composition with the augmenation
$\epsilon$
is trivial homomorphism
$C^{*}[\pi ]\mapr{}{\bf C}$
generated by the homomorphism of the group
$\pi $
into trivial group
$e$.
Then its restriction to the group
$\pi \subset C^{*}[\pi ]$
is generated by an asymptotic representation of the group
$\pi $.
\medskip
In a similar way one can define algebras
\begin{displaymath}\label{eq(1)26}
Q^{+}_{\alpha}(\hbox{Comp}(H))=
B^{+}_{\alpha}(\hbox{Comp}(H))/I,
\end{displaymath}
where $\hbox{Comp}$
is the $C^*$-algebra of compact operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$K$-groups of the asymptotic $C^*$-algebra}\label{subsec1.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we calculate the $K$-groups of the $C^*$-algebras
$B^{(+)}_{\alpha }(M_{f})$ and $B^{(+)}_{\alpha }(\hbox{Comp}(H))$. As the
calculation is the same for both cases, we will restrict ourselves to the
firts case. For shortness sake put
\begin{displaymath}\label{eq(1)31}
\begin{array}{ccc}
B^{+}_{\alpha }&=& B^{+}_{\alpha }(M_{f}),\cr
B_{\alpha }&=& B_{\alpha }(M_{f});\cr
\end{array}
\end{displaymath}
\begin{displaymath}\label{eq(1)32}
\begin{array}{ccc}
Q^{+}_{\alpha }&=&Q^{+}_{\alpha }(M_{f}) ,
\cr
Q_{\alpha }&=&Q_{\alpha }(M_{f}).
\cr
\end{array}
\end{displaymath}
Let
${\bf e}=\{(e_k)\}\in B$
be a sequence of diagonal matrices which has the unit on the first place
and zeros otherwise
$e_k\in M_{n_k}$.
It is clear that
${\bf e}\in B_{\alpha }$
and it is a projection.
Another projection
${\bf 1}\in B^{+}_{\alpha } $, being
the unit of the algebra
$B^{+}_{\alpha }$,
can be represented as a sequence
$(1,\{p_{k}=0\})$.
\begin{lem}\label{lm(1)1}
One has
\begin{displaymath}\label{eq(1)33}
\begin{array}{ccl}
K_0(Q^+_\alpha) & \cong &
{\bf Z}\oplus{\bf Z} \hbox{ with the generators }
[{\bf e}] \hbox{ and }
[{\bf 1}]; \cr
K_1(Q^+_\alpha)&=&0.
\end{array}
\end{displaymath}
\end{lem}
{\bf Proof.}
Let us construct a homomorphism
\begin{equation}\label{eq(1)34}
\hbox{tr}=\hbox{tr}_{1}\oplus\hbox{tr}_{2}:
K_0(Q^+_\alpha)\mapr{}{\bf Z}\oplus{\bf Z}
\end{equation}
Let
$p\in M_r(Q^{+}_\alpha)$
be an arbitrary projection and let
$p'\in M_r(B^{+})$
be a projection which is the lift of
$p$.
The projection $p'$ exists. Indeed, let
\begin{eqnarray*}\label{eq(1)35}
p'=(\lambda ,\{p'_{k}\})\in M_r(B^{+})) ,\nonumber\\
p''= \{p'_{k}\}\in M_r(B),\nonumber \\
p'_{k}\in M_{r}\ll(M_{n_{k}}\rr)=M_{n_{k}}\ll(M_{r}({\bf C})\rr),
\nonumber\\
\lambda \in M_{r}({\bf C})
\end{eqnarray*}
be a lift of
$p\in M_r(Q^{+}_\alpha)$.
The condition
$p'\in(q^{+})^{-1}\ll(Q^+_{\alpha }\rr)=
B^+_{\alpha }$
means that
\begin{equation}\label{eq(1)36}
\|p'_{k}\oplus 0\cdot{\bf E}_{(n_{k+1}-n_{k})}-p'_{k+1}\|\mapr{}0.
\end{equation}
As
$p^{2}=p$ one has
$q\ll((p')^{2}-p'\rr)=0$.
Hence
\begin{displaymath}\label{eq(1)37}
\lim\|\ll((\lambda ,p'_{k})\rr)^{2}-(\lambda ,p'_{k})\|=0.
\end{displaymath}
Therefore
\begin{displaymath}\label{eq(1)38}
\lambda^{2}-\lambda =0,
\end{displaymath}
and
\begin{displaymath}\label{eq(1)39}
\lim\|\ll((\lambda {\bf E}_{n_{k}}+p'_{k})\rr)^{2}-
\ll(\lambda {\bf E}_{n_{k}}+p'_{k}\rr)\|=0.
\end{displaymath}
This means that starting from a number
$k$
the spectrum of the matrix
$\lambda {\bf E}_{n_{k}} +p'_{k}$
is uniformly with respect to $k$ concentrated in a small neighborhood
$D_{k}$
of zero and unit,
$D_{k}=D^{0}_{k}\cup D^{1}_{k}$, and
\begin{displaymath}\label{eq(1)40}
\hbox{diam}\ll(D^{i}_{k}\rr)\mapr{}0.
\end{displaymath}
Put
\begin{equation}\label{eq(1)41}
\tilde p'_{k}=
\int_{\gamma }\ll(\lambda{\bf E}_{n_{k}} +p'_{k}-\mu \rr)^{-1}d\mu -
\lambda{\bf E}_{n_{k}},
\end{equation}
where
$\gamma $
is a closed curve enveloping domains
$D^{1}_{k}$
and not intersecting with
$D^{0}_{k}$.
It is clear that the matrix
$\tilde p'_{k}+\lambda {\bf E}_{n_{k}}$,
defined by the formula
(\ref{eq(1)41}),
is a projection for which one has
\begin{equation}\label{eq(1)42}
\|\tilde p'_{k}- p'_{k}\|\mapr{}0.
\end{equation}
Hence the sequence
\begin{displaymath}\label{eq(1)43}
\tilde p'=
(\lambda ,\{\tilde p'_{k} \})
\end{displaymath}
also is projection which is the lift of the element
$p$.
Put
\begin{equation}\label{eq(1)44}
\hbox{tr}_{1}(p) = \hbox{tr}(\lambda )\in {\bf Z},
\end{equation}
\begin{equation}\label{eq(1)45}
\hbox{tr}_{2}(p) =\lim\limits_{k\mapr{}\infty}
\hbox{tr}(p'_{k}).
\end{equation}
The limit in the
(\ref{eq(1)45})
exists since due to the condition
(\ref{eq(1)36})
one has
\begin{displaymath}\label{eq(1)46}
\lim\limits_{k\mapr{}\infty}
\ll(\hbox{tr}(p'_{k})-\hbox{tr}(p'_{k+1})\rr)=0.
\end{displaymath}
On the other side due to
(\ref{eq(1)41})
one has
\begin{equation}\label{eq(1)47}
\|\tilde p'_{k+1}-
\tilde p'_{k}\oplus 0\cdot{\bf E}_{(n_{k+1}-n_{k})}\|\mapr{}0,
\end{equation}
that is
\begin{equation}\label{eq(1)48}
\hbox{tr}\ll(
\tilde p'_{k+1}\rr)- \hbox{tr}\ll(\tilde p'_{k}\rr)
\mapr{}0.
\end{equation}
As the matrix
$\tilde p'_{k}$
is a difference of two projections, so the trace has an integer value.
Hence existence of the limit
$\lim\hbox{tr}\ll(\tilde p'_{k}\rr)$ follows from (\ref{eq(1)48}), so we
get existence of the limit in the formula (\ref{eq(1)45}).
Notice that the definitions
(\ref{eq(1)44})
and
(\ref{eq(1)45})
are correct,
that is they do not depend on
choice of the projection
$p$
which represents the element of the
$K$-group, and do not depend on choice
of lifting projection
$\tilde p'$.
\medskip
Let show that the homomomorphism
(\ref{eq(1)34})
is isomorphism.
Notice that starting from a
$k$ the projectors
$\tilde p'_{k+1}+\lambda{\bf E}_{n_{k+1}}$
and
$\ll(\tilde p'_{k}+\lambda{\bf E}_{n_{k}}\rr)\oplus
\lambda{\bf E}_{(n_{k+1}-n_{k})}$
are equivalent that is
\begin{displaymath}\label{eq(1)49}
\tilde p'_{k+1}+\lambda{\bf E}_{n_{k+1}}=
U_{k+1}^{-1}\ll(\ll(\tilde p'_{k}+\lambda{\bf E}_{n_{k}}\rr)
\oplus\lambda{\bf E}_{(n_{k+1}-n_{k})}\rr)U_{k+1},
\end{displaymath}
and also the unitary matrices
$U_{k}$ can be chosen converged to the unit operator:
\begin{displaymath}\label{eq(1)50}
\|U_{k}-{\bf E}\|\mapr{}0.
\end{displaymath}
Passing to stabilization one has
\begin{eqnarray*}\label{eq(1)51}
\lefteqn{
\ll(\tilde p'_{k+1}+\lambda{\bf E}_{n_{k+1}}\rr)
\oplus\lambda{\bf E}_{(n_{k+s}-n_{k+1})}
}\nonumber\\
&&
=\ll(U_{k+1}\oplus{\bf E}_{(n_{k+s}-n_{k+1})}\rr)^{-1}
\nonumber\\&&\hskip 2cm\cdot
\ll(\ll(\tilde p'_{k}+\lambda{\bf E}_{n_{k}}\rr)
\oplus\lambda{\bf E}_{(n_{k+s}-n_{k})}\rr)
\ll(U_{k+1}\oplus{\bf E}_{(n_{k+s}-n_{k+1})}\rr),
\end{eqnarray*}
Hence
\begin{eqnarray*}\label{eq(1)52}
\lefteqn{
\tilde p'_{k+s}+\lambda{\bf E}_{n_{k+s}}}
\nonumber \\&&=
U_{k+s}^{-1}\ll(U_{k+s-1}\oplus{\bf E}_{(n_{k+s}-n_{k+s-1})}\rr)^{-1}
\cdots \ll(U_{k+1}\oplus{\bf E}_{(n_{k+s}-n_{k+1})}\rr)^{-1}
\nonumber \\ &&\cdot
\ll(\ll(\tilde p'_{k}+\lambda{\bf E}_{n_{k}}\rr)\oplus\lambda{\bf E}_{(n_{k+s}-n_{k})}\rr)
\nonumber\\&&\cdot
\ll(U_{k+1}\oplus{\bf E}_{(n_{k+s}-n_{k+1})}\rr)\cdots
\ll(U_{k+s-1}\oplus{\bf E}_{(n_{k+s}-n_{k+s-1})}\rr)
U_{k+s}
\nonumber\\&&=
V_{k+s}^{-1}
\ll(\ll(\tilde p'_{k}+\lambda{\bf E}_{n_{k}}\rr)\oplus\lambda{\bf E}_{(n_{k+2}-n_{k})}\rr)
V_{k+s}.
\end{eqnarray*}
The sequence
$V=\{V_{k+s}\}$
belongs to the algebra
$B_{\alpha}$ since
\begin{displaymath}\label{eq(1)53}
V_{k+s+1}-V_{k+s}=V_{k+s}\ll(U_{k+s+1}-1\rr)\mapr{}0,\quad s\mapr{}\infty.
\end{displaymath}
Hence
\begin{displaymath}\label{eq(1)54}
\ll(1,\{V_{k+s}-{\bf E}_{n_{k+s}}\}\rr)\in B^{+}_{\alpha }
\end{displaymath}
Thus the projection
$\tilde p'$
is equivalent to the projection
$s=\ll(\lambda ,\{s_{k+s}\}\rr)$. The latter has identical terms:
\begin{displaymath}\label{eq(1)55}
s_{k+s}=
\ll(\tilde p'_{k}\oplus 0\cdot{\bf E}_{(n_{k+s}-n_{k})}\rr).
\end{displaymath}
Therefore the projection
$s$ equals to a direct sum of summands which
are equivalent to the projection
${\bf e}$
or to the projection
${\bf 1}$.
Thus as
% \begin{eqnarray*}\label{eq(1)56}
$$
\hbox{tr}_{1}({\bf 1}) = 1 , \qquad
\hbox{tr}_{2}({\bf 1}) = 0,
$$
% \end{eqnarray*}
% \begin{eqnarray*}\label{eq(1)57}
$$
\hbox{tr}_{1}({\bf e}) = e , \qquad
\hbox{tr}_{2}({\bf 1}) = 1,
$$
% \end{eqnarray*}
one has both surjectivity and injectivity of the homomorphism
(\ref{eq(1)34}).
\medskip
Let pass to the calculation of the group
$K_1(Q^+_\alpha)=0$.
One should show that the group of all invertible elements
in the the matrix algebra
$M_{r}\ll(Q^{+}_{\alpha}\rr)$ has only a component.
Let
$u=(\lambda ,\{u_{k}\})\in M_{r}\ll(Q^{+}_{\alpha}\rr)$
be an invertible element. Then in reality
$\lambda \in M_{r}({\bf C})$ is invertible matrix.
Hence there is a homotopy to the case when
$\lambda ={\bf E}_{r}$.
Indeed, let
$\lambda_{t}$,
\begin{displaymath}\label{eq(1)58}
\begin{array}{ccc}
\det\lambda_{t}\neq 0, &0\leq t\leq 1,&\cr
\lambda_{0}=\lambda ,&\lambda_{1}={\bf E}_{r},&
\end{array}
\end{displaymath}
be a homotopy of the matrix
$\lambda $.
Then put
\begin{displaymath}\label{eq(1)59}
u_{t}=\ll(\lambda_{t},
\ll\{\ll(
\lambda_{t}\otimes{\bf E}_{n_{k}}
\rr)
\ll(
\lambda_{0}\otimes{\bf E}_{n_{k}}
\rr)^{-1}u_{k} \rr\}
\rr)=
\ll(\lambda_{t},
\{\ll(\lambda_{t}\lambda_{0}^{-1}\otimes{\bf E}_{n_{k}}\rr)u_{k}\}\rr).
\end{displaymath}
It is easy to check that
\begin{displaymath}\label{eq(1)60}
u_{t}\in M_{r}\ll(B^{+}_{\alpha }\rr)
\end{displaymath}
is continuous with respect to
$t$
and invertible.
Put
\begin{displaymath}\label{eq(1)61}
u_{1}=\ll({\bf E}_{r},\{\ll(\lambda_{0}^{-1}\otimes{\bf E}_{n_{k}}\rr)u_{k}\} \rr)=
\ll({\bf E}_{r}, \{v_{k}\}\rr),
\end{displaymath}
where
\begin{displaymath}\label{eq(1)62}
v_{k}=\ll(\lambda_{0}^{-1}\otimes{\bf E}_{n_{k}}\rr)u_{k}.
\end{displaymath}
Let extend the definition of the function
$v(t)$
for all real values of the variable
$1\geq t <\infty$
taking the convex linear combination on each interval
$[k,k+1]$:
if
$t=k+\tau , 0<\tau <1$,
then put
\begin{displaymath}\label{eq(1)63}
v_{t}=\tau v_{k+1}+(1-\tau )v_{k}.
\end{displaymath}
Requied homotopy of the element
$u_{1}$
one can construct by the formula
\begin{displaymath}\label{eq(1)64}
u_{t}=\ll({\bf E}_{r}, \{v_{k,t}\}\rr), 1\geq t \geq 2,
\end{displaymath}
where
\begin{displaymath}\label{eq(1)65}
v_{k,t}=v((t-1)k).
\end{displaymath}
\q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic representations as
representations into the Calkin algebra}\label{subsec1.4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Denote by
${\cal Q}$
the Calkin algebra
\begin{displaymath}\label{eq(1)66}
{\cal Q}(H)=B(H)/\hbox{Comp}(H),
\end{displaymath}
where
$H$
is Hilbert space,
$B(H)$
is the bounded operator algebra,
$\hbox{Comp}(H)$
is the subalgebra of compact operators.
\begin{thm}\label{hom}
There is a natural homomorphism
\begin{equation}\label{eq(1)67}
\psi:Q^+_\alpha(\hbox{\rm Comp})\otimes C(S^1)\arr{\cal Q}(H),
\end{equation}
such that the induced homomorphism
\begin{equation}\label{eq(1)68}
\psi_*:K_1(Q_\alpha(\hbox{\rm Comp})\otimes C(S^1))\arr K_1({\cal Q})
\end{equation}
is isomorphism.
\end{thm}
{\bf Proof.}
Let us start from the definition of the homomorphism
(\ref{eq(1)67}).
Let represent the Hilbert space
$H$
as a direct infinite sum of its copies:
\begin{displaymath}\label{eq(1)69}
\tilde H = \bigoplus\limits_{k=-\infty}^{\infty}H_{k},\quad H_{k}=H.
\end{displaymath}
The homomorphism
$\psi $ will be conctructed as a homomorphism
into the Calkin algebra
${\cal Q}(\tilde H)$.
Consider an element
$v\in Q^+_\alpha(\hbox{Comp})$,
and let
$v'=(\lambda ,\{v'_k\})\in B^+_{\alpha }(\hbox{Comp})$
be a lift of the element
$v$.
Define the operator
$\psi(v)$
which acts on the space
$\tilde H$
as follows: if $k\geq 1$, then the operator
$\psi (v)$ acts on the subspace
$H_k$
as
$\lambda +v'_k$;
if $k\leq 0$ then the operator $\psi (v)$
acts on the subspace
$H_{k}$
as multiplication with the number
$\lambda$.
\medskip
The operator $\psi (v)$ is not uniquely defined, but only to a
compact summand. Hence $\psi (v)$ defines correctly an element
$\bar\psi(v)\in{\cal Q}(\tilde H)$.
Put
$\psi(v\otimes 1_{C(S^1)})=\bar\psi(v)$.
It remains to define the value of $\psi $ on the function
$u=e^{2\pi i t}\in C(S^1)$.
Consider the Fredholm operator
$F$
(of the index $0$) on the Hilbert space
$\tilde H$
as the shift operator with respect to the number
$k$:
\begin{displaymath}\label{eq(1)70}
F:H_{k+1}\mapr{\equiv}H_{k}.
\end{displaymath}
The operator
$F$
commutes with the image
$\psi(Q^+_\alpha(\hbox{Comp}))$
up to compact operators. Hence one has correctly defined
homomorphism
$\psi$
putting
\begin{displaymath}\label{eq(1)71}
\psi(1_{Q^+_\alpha}\otimes e^{2\pi i t})=F.
\end{displaymath}
Let prove that the homomorphism
(\ref{eq(1)68})
is isomorphic.
As
$K_1(Q^+_\alpha(\hbox{Comp}))=0$,
due to the K\"unneth formula one has:
\begin{displaymath}\label{eq(1)72}
K_1(Q^+_\alpha\otimes C(S^1))\cong K_0(Q^+_\alpha)\otimes K^1(S^1).
\end{displaymath}
\medskip
Let
$[u]\in K^1(S^1)$
be a generator. Then the group
$K_1(Q^+_\alpha\otimes C(S^1))$
is generated by two elements:
$[e]\otimes [u]$
and
$[1]\otimes [u]$.
As
\begin{displaymath}\label{eq(1)73}
\psi(1\otimes u)=F\quad {\rm and}\quad {\rm ind}\, F=0,
\end{displaymath}
ane has
\begin{displaymath}\label{eq(1)74}
\psi_*([1]\otimes [u])=0.
\end{displaymath}
Let us calculate the element
$\psi_*([e]\otimes [u])$.
Due to
\cite{connes}
one has
\begin{displaymath}\label{eq(1)74.1}
[e]\otimes [u]=[(1-e)\otimes 1+e\otimes u],
\end{displaymath}
hence
\begin{eqnarray*}\label{eq(1)75}
\psi_*([e]\otimes [u])
&=&
\psi_*([(1-e)\otimes 1+e\otimes u])
\nonumber\\
&=&
[1-\psi(e\otimes 1)+\psi(e\otimes u)]=
[1-\psi(e)+\psi(e)F].
\end{eqnarray*}
As
$e\in Q_{\alpha }$
is a projection and as
$\psi (e)$
commutes with
$F$
up to compact operators, so one has
\begin{equation}\label{eq(1)76}
\psi_{*}([e]\otimes [u])=[1-\psi(e)+\psi (e)F\psi (e)].
\end{equation}
Thus the operator
$G$
in the right side of
(\ref{eq(1)76})
is a direct sum of two operators:
one of them is identical on the image of the projection
$1-\psi(e)$, the second is the restriction
$F'$
of the operator
$F$
on the image of the projection
$\psi(e)$.
It is easy to check that the index of the operator $F'$ equals
to one. Indeed, the image of the projection
$\psi(e)$ is decomposed into a direct sum of
one dimensional subspaces generated by the first basis vectors
$e_{1,k}$
in the Hilbert spaces
$H_{k}$
for
$k\geq 1$.
On the other side the operator
$F'$
is a shift which
sends the vector
$e_{1,k}$
to the vector
$e_{1,k-1}$.
This means that the ${\rm ind}\, F'=1$.
Thus
${\rm ind}\, G=1$,
hence the homomorphism
$\psi_*$
sends the generator of the group
$K_1(Q_\alpha\otimes C(S^1))\cong{\bf Z}$
into the generator of the group
$K_1({\cal Q})$. \q
\medskip
Denote by
${\cal R}_{asym}(\pi)$
the Grothendieck group of virtual asymptotic representations
of the group
$\pi$.
Let
$\til{\cal R}_{asym}(\pi)$
be the kernel of the map
\begin{equation}\label{eq(1)76.1}
{\cal R}_{asym}(\pi)\arr {\cal R}_{asym}(e)\cong {\bf Z},
\end{equation}
which is defined by the restriction of the representation
on trivial subgroup
($e$).
Let
$B\pi$
be the classifying space of the group
$\pi$.
Remind that in section \ref{construction}
a homomorphism
\begin{equation}\label{eq(1)77}
\phi:\widetilde{\cal R}_{asym}(\pi)\arr K^0(B\pi),
\end{equation}
was defined, which can be described as follows.
Let
$C^*[\pi]$
be the group
$C^*$-algebra of the group
$\pi$,
and let
$\xi\in K^0_{C^*[\pi]}(B\pi)$
be the universal bundle.
The asymptotic representation
$\sigma$
defines a homomorphism
\begin{equation}\label{eq(1)78}
\ov{\sigma}:C^*[\pi]\arr Q^+_\alpha,
\end{equation}
which maps the universal bundle
$\xi$
to an element
$\ov{\sigma}_*(\xi)\in K^0_{Q^+_\alpha}(B\pi)$.
The homomorphism
(\ref{eq(1)24})
defines a homomorphism
\begin{displaymath}\label{eq(1)79}
\phi':{\cal R}_{asym}(\pi)\arr K^0_{Q^+_\alpha}(B\pi)
\end{displaymath}
by the formula
\begin{displaymath}\label{eq(1)80}
\phi'(\sigma)=\ov{\sigma}_{*}(\xi).
\end{displaymath}
Consider a commutative diagram
\begin{displaymath}\label{eq(1)81}
\begin{array}{ccccc}
\til{\cal R}_{asym}(\pi)&\arr&{\cal R}_{asym}(\pi)&
\arr&{\cal R}_{asym}(e)\\
\downarrow&&\downarrow\lefteqn{\phi'}&&\downarrow\lefteqn{\phi'_e}\\
K^0_{Q_\alpha}(B\pi)&\arr&K^0_{Q^+_\alpha}(B\pi)&
\arr&K^0_{\bf C}(B\pi)
\end{array}
\end{displaymath}
It follows from the lemma \ref{lm(1)1}, from the K\"unneth formula and from
(\ref{eq(1)20}) that the lower line of the diagram
is exact, therefore the left vertical arrow is well-defined.
As we have
$$
K^0_{Q_\alpha}(B\pi)=K^0(B\pi)\otimes K_0(Q_\alpha)\cong K^0(B\pi),
$$
so we can define $\phi$ to be this left vertical arrow after identifying
$K^0_{Q_\alpha}(B\pi)$ with $K^0(B\pi)$.
Notice that the image of the
homomorphism $\phi'_e$ coincides with the subgroup in
$K^0(B\pi)$ generated by the trivial representations.
\medskip
As there exists a natural isomorphism
$$
j:K^0_A(B\pi\times S^1\times S^1)\widetilde{\arr}
K^0_{A\otimes C(S^1)}(B\pi\times S^1)
$$
for any $C^*$-algebra $A$, so multiplication by the Bott generator
$\beta\in K^0(S^1\times S^1)$ defines an inclusion
\begin{equation}\label{eq(1)82}
\begin{array}{ccccl}
\ov{\beta}_A:K^0_A(B\pi)&\stackrel{\otimes\beta}{\arr}&
K^0_A(B\pi\times S^1\times S^1)&\stackrel{j}{\arr}&
K^0_{A\otimes C(S^1)}(B\pi\times S^1)\\
&&&=&K^0_{A\otimes C(S^1)}
(B(\pi\times{\bf Z})).
\end{array}
\end{equation}
In the case $A=Q_\alpha$ we will write $\ov{\beta}$ instead of
$\ov{\beta}_{Q_\alpha}$.
\medskip
Now denote by ${\cal R}_{\cal Q}(\pi)$ the group of (virtual)
representations of the group $\pi$ into the Calkin algebra.
It is easily seen that the homomorphism (\ref{eq(1)77}) allows us to define
a homomorphism
\begin{equation}\label{eq(1)83}
\ov{\psi}:{\cal R}_{asym}(\pi)\arr {\cal R}_{\cal Q}(\pi\times
{\bf Z})
\end{equation}
given by the formula $\ov{\psi}(\ov{\sigma})=\psi(\ov{\sigma}\otimes id)$
for $\ov{\sigma}\in{\cal R}_{asym}(\pi)$ and
$$
\ov{\sigma}\otimes id: C^*[\pi\times {\bf Z}]\cong
C^*[\pi]\otimes C(S^1)\arr
Q^+_\alpha\otimes C(S^1).
$$
Let $\eta\in K^0_{C^*[\pi\times{\bf Z}]}(B(\pi\times Z))$
be the universal bundle over $B(\pi\times{\bf Z}$.
Then there exists also a homomorphism
\begin{equation}\label{eq(1)84}
f:{\cal R}_{\cal Q}(\pi\times{\bf Z})\arr
K^0_{\cal Q}(B(\pi\times{\bf Z}))
\end{equation}
defined as the image of $\eta$ over
$B(\pi\times{\bf Z})$ under the representations into the Calkin algebra.
\medskip
All these homomorphisms (\ref{eq(1)68}) -- (\ref{eq(1)83})
can be represented by the diagram
\begin{equation}\label{eq(1)85}
\begin{array}{ccccc}
\widetilde{\cal R}_{asym}(\pi)&\stackrel{\phi}{\arr}&
K^0_{Q_\alpha}(B\pi)&\stackrel{\ov{\beta}}{\arr}&
K^0_{Q_\alpha\otimes C(S^1)}(B\pi\times S^1)\\
\downarrow&&&&\downarrow\lefteqn{\psi_*}\\
{\cal R}_{asym}(\pi)&\stackrel{\ov{\psi}}{\arr}&
{\cal R}_{\cal Q}(\pi\times{\bf Z})&\stackrel{f}{\arr}&
K^0_{\cal Q}(B\pi\times S^1),
\end{array}
\end{equation}
where the left vertical arrow is the inclusion.
\begin{thm}\label{diagram}
The diagram $($\ref{eq(1)85}$)$ is commutative.
\end{thm}
{\bf Proof.} Direct calculation. \q
\medskip
So now we have the homomorphism
\begin{equation}\label{eq(1)86}
\ov{\phi}:\widetilde{\cal R}_{asym}(\pi)\arr
K^0_{\cal Q}(B\pi\times S^1)\cong K^1(B\pi\times S^1).
\end{equation}
\medskip
Theorem \ref{diagram} shows that every element of the group $K^0(B\pi)$
which can be obtained by an
asymptotic representation of the fundamental group $\pi$ can be obtained
also by a
representation of the group $\pi\times {\bf Z}$ into the Calkin algebra.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic representations versus representations into the
Calkin algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we are going to show that there is much less asymptotic
representations than representations into the Calkin algebra.
To do so we have to return to the example of the group $\G$ of the
subsection \ref{example_of_a_group}. We show that
the element $\b\in K^0(B\G)$ (\ref{beta}) can be obtained as
an image of some representation of the group $\G\times{\bf Z}$
into the Calkin algebra under the mapping $\alpha$ (\ref{sborka}).
Remember that in section \ref{construction} a map
\begin{equation}\label{mapasym}
{\cal R}_{asym}(\G)\arr K^0(B\G)
\end{equation}
was constructed, which factorizes (see subsection \ref{subsec1.4})
through the map
\be\label{sborka}
\alpha:{\cal R}_{\cal Q}(\G\times{\bf Z})\arr K^0(B\G),
\ee
defined by the universal vector bundle (a map dual to the assembly map),
where ${\cal R}_{\cal Q}(\G\times{\bf Z})$ denotes the Grothendieck group of
representations of $\G\times{\bf Z}$ into the Calkin algebra ${\cal Q}$.
It follows from the Corollary \ref{RG=RH} that ${\cal R}_{asym}(\G)={\cal
R}_{asym}(H)$, therefore the element $\beta\in K^0(B\G)$ does not lie in
the image of the map (\ref{mapasym}), hence we obtain
\begin{cor}
The map (\ref{mapasym}) is not a rational epimorphism for the group $\G$.
\q
\end{cor}
\begin{prop}
There exists a representation $\rho\in {\cal R}_{\cal
Q}(\G\times{\bf Z})$ such that $\alpha(\rho)=\beta\in K^0(B\G)$.
\end{prop}
{\bf Proof.}
Let us describe how to construct such $\rho$.
Denote by $I_2$ the $2{\times 2}$ unit matrix, and let $J_2$ be the
matrix $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$. Consider the
sequence $\{\s_n\}$ of almost representations of the group
$\G$ mentioned in the proof of Theorem \ref{primergruppy} and put for
$n\geq 3$
$$
\s'_n(a)=\s_{n-2}(a)\oplus I_2, \quad
\s'_n(c)=\s_{n-2}(c)\oplus I_2, \quad
\s'_n(b)=\s_{n-2}(b)\oplus J_2.
$$
\begin{lem}\label{put}
There exists a homotopy $\{\s_n^t\}_{t\in [0,1]}$ of almost
representations of the group $\G$ connecting $\s_n$ with $\s'_n$ such that
$\nnn{\s_n^t}\leq 3\e_n=3 |e^{2\pi i/n}-1|$.
\end{lem}
{\bf Proof.}
Let $e_1,\ldots,e_n$ be a basis in ${\bf C}^n$ such that the almost
representation $\s_n$ has the above form. Let $u\in M_n({\bf C})$
be the unitary matrix, which transposes the vectors $e_1$
and $e_{n-1}$ and does not move the other vectors of the basis, let
$u_t\in M_n({\bf C})$ be a unitary path connnecting the unit matrix
with the matrix $u$ and such that it does not move these other basis
vectors. Then the path $\s_n^t(b)=u_t^*\s_n(b)u_t$ connects the
matrices $\s_n(b)$ and $\s'_n(b)$. One can directly check that
$\norm{(\s_n^t(b))^2-I}\leq\e_n$ and
$\norm{(\s'_n(a)\s_n^t(b))^2-I}\leq 3\e_n$. It is shown in \cite{mish-noor}
that the Voiculescu pair of matrices $\s_n(a)$ and $\s_n(c)$ can be
connected with the matrices $\s'_n(a)$ and $\s'_n(c)$ by a path such that
$\norm{\s_n^t(a)\s_n^t(c)\s_n^t(a)^{-1}\s_n^t(c)^{-1}-I}\leq \e_n$.
The proof is completed by a direct check of the estimate
$\norm{(\s_n^t(a)\s_n(b))^2-I}\leq \e_n$. \q
We proceed with the construction of the representation $\rho$.
On a Hilbert space $H$ consider the operators given by the matrices
$$
\ov{\s}_n(a)=\s_n(a)\oplus I_2\oplus\ldots\oplus I_2\oplus\ldots,\quad
\ov{\s}_n(c)=\s_n(c)\oplus I_2\oplus\ldots\oplus I_2\oplus\ldots,\quad
$$
$$
\ov{\s}_n(b)=\s_n(b)\oplus J_2\oplus\ldots\oplus J_2\oplus\ldots.
$$
Denote by $F$ the set $\{a,b,c\}$ of generators of the group $\G$.
By Lemma \ref{put} the sequences of operators
$\ov{\s}_n(g)$, $g\in F$, can be included into continuous paths
$\ov{\s}_t(g)$, $t\in [1,\infty)$, such that for
$t\geq n$ all relations of the group $\G$ are satisfied up to
$3\e_n$. Then as in \cite{mish-noor} one can choose an increasing
sequence $\{t_k\}$ of the values for the parameter
$t$ so that the operators $\ov{\s}_{t_k}(g)=\ov{\s}_k(g)$ would satisfy
$$
\lim_{k\to\infty}\norm{\ov{\s}_{k+1}(g)-\ov{\s}_k(g)}=0, \quad g\in F,
$$
$$
\lim_{k\to\infty}\norm{r(\ov{\s}_k(a),\ov{\s}_k(b),\ov{\s}_k(c))-I}=0,
$$
for every relation $r=r(a,b,c)$ of the group $\G$.
Without loss of generality one can assume that the sequence of operators
$\{\ov{\s}_k(g)\}$ contains $\{\ov{\s}_n(g)\}$ as a subsequence.
Put ${\cal H}=\oplus_k H_k$, where $H_k=H$, and for negative $k$
put $\ov{\s}_k(g)=I$, $g\in F$.
Let $T$ be the shift in
${\cal H}$, $T(H_k)=H_{k-1}$. Denote by $d$ a generator for the group
${\bf Z}$. Define the representation $\rho$ on the generators of the
group $\G\times{\bf Z}$ by
$$
\rho(g)=\oplus_k\ov{\s}_k(g),\quad g\in F;\qquad
\rho(d)=T.
$$
It is easy to check that all relations of the group $\G\times{\bf Z}$
are satisfied in the Hilbert space ${\cal H}$ modulo compact operators,
hence $\rho$ is well defined as a representation into the Calkin algebra
(cf. \cite{man-mish}).
As the restriction of this representation onto the subgroup
${\bf Z}^2\times{\bf Z}\subset \G\times{\bf Z}$ is obtained from the
Voiculescu matrices, so this representation defines the Bott generator in
the group $\til{K}^0(B{\bf Z}^2)$, hence the representation
$\rho$ of the group $\G\times{\bf Z}$ into the Calkin algebra defines the
element $\beta\in K^0(B\G)$. \q
\medskip
Remark that in our example the absence of asymptotic stability is related
to torsion. It would be interesting to know whether torsion-free groups
always are asymptotically stable.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymptotic representations as generalized Fredholm
representations}\label{sec2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Notice that the group $K^0_{\cal Q}(B\pi\times S^1)$ can be decomposed
into direct sum:
\begin{equation}\label{eq(2)87}
K^0_{\cal Q}(B\pi\times S^1)=K^0_{\cal Q}(B\pi)\oplus
K^0_{\cal Q}(B\pi\wedge S^1)\cong K^1(B\pi)\oplus K^0(B\pi)
\end{equation}
induced by an inclusion map $i:s_0\arr S^1$, where $s_0\in S^1$,
and the image of the homomorphism $\ov{\phi}$ (\ref{eq(1)86}) lies only
in the second summand of (\ref{eq(2)87}). Indeed, consider the composition
of the map $\ov{\phi}$ with the map $i^*:K^0_{\cal
Q}(B\pi\times S^1)\arr K^0_{\cal Q}(B\pi)$. But as the
multiplication by the Bott generator is involved in the map $\ov{\phi}$,
so its composition with $i^*$ gives the zero map. Therefore the image of
the group $\widetilde{\cal R}_{asym}(\pi)$ lies in $K^0(B\pi)$ and hence
defines a (virtual) vector bundle over $B\pi$.
\medskip
On the other hand the image of the map (\ref{eq(1)84}) need not be
contained in the second summand of (\ref{eq(2)87}). It would be so if the
representation of the group $\pi\times {\bf Z}$ into the Calkin algebra
would be a part of a Fredholm representation of the group $\pi$
\cite{mish}. Using the notion of asymptotic representations we can now give
a generalization of the Fredholm representations which would also ensure
that the image of such representations would lie in the second summand of
(\ref{eq(2)87}).
Переделать с F на образующие
\medskip
Let $\rho:\pi\times {\bf Z}\arr {\cal Q}$ be a representation
into the Calkin algebra and let $F\subset\pi$ denote a finite subset.
Denote by $B(H)$ the algebra of bounded
operators on a separable Hilbert space $H$ and let $q:B(H)\arr
{\cal Q}$ be the canonical projection.
\begin{dfn}
{\rm
We call a map
$\tau:\pi\arr B(H)$ an {\em $\e$-trivialization} for $\rho$
if
\begin{enumerate}
\item
$\|\tau(gh)-\tau(g)\tau(h)\|\leq\e$ for any $g,h\in F\subset\pi$,
\item
$q(\tau(g))=\rho(g;0)$ for any $g\in\pi$, $(g;0)\in\pi\times{\bf Z}$.
\end{enumerate}
}
\end{dfn}
\begin{dfn}
{\rm
Suppose that for every finite $F\subset\pi$ there
exists for every $\e>0$ an $\e$-trivialization $\tau_\e$ for $\rho$. Then the pair
$(\tau_\e,\rho)$ is called an {\em asymptotic Fredholm representation}.
}
\end{dfn}
Let $u$ be a generator of the group $\bf Z$.
Notice that the image of the group $\pi$ under
$\varepsilon$-trivializations
commutes with
some Fredholm operator $T=\rho(0,u)$ modulo
compacts.
Denote the group of all asymptotic Fredholm representations by ${\cal
R}_{aF}(\pi)\subset {\cal R}_{\cal Q}(\pi\times{\bf Z})$.
\begin{prop}
The image of ${\cal R}_{aF}$ under the map $f$ $($\ref{eq(1)84}$)$
lies in the group $K^0(B\pi\wedge S^1)$.
\end{prop}
{\bf Proof.}
It was described in the section \ref{construction} how to construct a bundle
over $B\pi$ with the fibers isomorphic to the Hilbert space $H$ and with
the structural group $GL(H)$
starting from an almost representation $\tau_\e$ for small
enough $\e$.
To do so one should construct transition functions acting on the fibers
$\xi_x$,
$$
T_g(x):\xi_x\arr\xi_{gx}
$$
for $g\in\pi$, $x\in E\pi$.
One should chose representatives $\{a_\a\}$ in each orbit of the set of
vertices of $E\pi$ and define $T_g(a_\a)=\tau_\e(g)$. Take now an
arbitrary vertex $b\in E\pi$. Then there exists such $h\in \pi$ that
$b=h(a_\a)$ and we should put $T_g(b)=\tau_\e(gh)\tau_\e^{-1}(h)$.
Further these transition functions should be extended by linearity to all
simplexes of $E\pi$.
But obviously
$$
q(T_g(b))=q(\tau_\e(gh)\tau_\e^{-1}(h))=q(\tau_\e(g))=\rho(g,0),
$$
hence after we pass to quotients,
the transition functions $q(T_g(x))=\rho(g,0)$
would become constant and the bundle with the
structural group being the invertibles of the Calkin algebra would be
locally flat. But as any
bundle with fibers $H$ is trivial, so the corresponding
quotient bundle with fibers
isomorphic to the Calkin algebra is trivial too and the projection of
$\varphi({\cal R}_{aF})$ onto the first summand of $K^0_{\cal Q}(B\pi)$ is
equal to zero. \q
\medskip
{\bf Remark.}
In the subsection \ref{subsec1.4} we have seen that an asymptotic
representation $\sigma=(\sigma_n)$ defines a homomorphism
$\rho:\pi\times{\bf Z}\arr GL({\cal Q})$ into the group of
invertibles of the Calkin algebra. But the same asymptotic representation
gives $\e$-trivializations of $\rho$ for any $\e>0$. Indeed we can define
them as the image in $B(H)$ of the almost representations
$(1,1,\ldots,1,\sigma_n(g),\sigma_{n+1}(g),\ldots)$, i.e.
$$
\tau_{\e_n}(g)={\bf E}\oplus{\bf E}\oplus\ldots\oplus{\bf E}\oplus
\sigma_n(g)\oplus\sigma_{n+1}(g)\oplus\ldots,
\qquad g\in\pi.
$$
So asymptotic representations define asymptotic Fredholm representations
and we have an inclusion ${\cal R}_{asym}(\pi)\subset{\cal R}_{aF}(\pi)$.
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%\end{document}
\vspace{1.5cm}
\noindent
\parbox{7cm}{%
V.~M.~Manuilov\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119899, RUSSIA\\
e-mail: manuilov@mech.math.msu.su
}
\hfill
\parbox{7cm}{%
A.~S.~Mishchenko\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119899, RUSSIA\\
e-mail: asmish@mech.math.msu.su
}
\end{document}