Obstructions to matricial stability of discrete groups and almost flat K-theory

TL;DR

This paper investigates the relationship between the stability of finite-dimensional representations of discrete groups and the vanishing of their rational cohomology, revealing obstructions to stability via K-theory classes.

Contribution

It establishes that for many groups, matricial stability implies vanishing rational cohomology in all nonzero even dimensions, and links almost flat K-theory classes to obstructions in stability.

Findings

Matricial stability implies vanishing rational cohomology in certain groups.

Almost flat K-theory classes can obstruct matricial stability.

Construction methods involve the dual assembly map and quasidiagonality.

Abstract

A discrete countable group G is matricially stable if the finite dimensional approximate unitary representations of G are perturbable to genuine representations in the point-norm topology. For large classes of groups G, we show that matricial stability implies the vanishing of the rational cohomology of G in all nonzero even dimensions. We revisit a method of constructing almost flat K-theory classes of BG which involves the dual assembly map and quasidiagonality properties of G. The existence of almost flat K-theory classes of BG which are not flat represents an obstruction to matricial stability of G due to continuity properties of the approximate monodromy correspondence.