On Almost Representations of Property (T) Groups

TL;DR

This paper extends the dichotomy of Property (T) to almost representations under a stronger condition, showing that either an almost invariant vector exists or vectors are far from invariant.

Contribution

It introduces a new dichotomy for almost representations of Property (T) groups under a strengthened condition, expanding the understanding of representation stability.

Findings

Dichotomy extends to almost representations under A.Zuk's condition

Almost invariant vectors either exist or are far from invariant in this setting

Strengthens the theoretical framework of Property (T) representations

Abstract

Property (T) for groups means a dichotomy: a representation either has an invariant vector or all vectors are far from being invariant. We show that, under a stronger condition of A.Zuk, a similar dichotomy holds for almost representations as well.