Almost representations of discrete groups and K-theory of classifying spaces
by
Vladimir Manuilov
Moscow State University

Let G be a finitely presented group. A mapping of a set of generators of G into the (infinite) unitary group is called an \epsilon-almost representation if the relations of G are satisfied up to \epsilon with respect to the operator norm. A continuous family of \epsilon_t-almost representations if lim_{t --> \infty}\epsilon_t=0. One of the reasons to study almost representations is their relation to the K-theory of classifying space of G.

If a group G is such that any its \epsilon-almost representation for small enough \epsilon generates an asymptotic representation then we call this group asymptotically stable (AS). We describe some classes of AS discrete groups. Obviously free groups are AS. Abelian groups and fundamental groups of oriented surfaces are also AS. The proof is based on some properties of almost commuting operators.

We also give an example of a group G without AS. This example is interesting also because it gives an example of a group without sufficient number of asymptotic representations. Namely elements of the K-functor of its classifying space cannot be obtained out of asymptotic representations.

Date received: June 2, 1999