Relative inner amenability

TL;DR

This paper introduces and explores the concept of relative inner amenability for subgroups within discrete groups, providing characterizations, examples, and implications for related properties like property gamma and Kazhdan's T.

Contribution

It defines the notion of relative inner amenability, establishes equivalent conditions, and connects it to properties of discrete groups such as property T and property gamma.

Findings

Characterization of relative inner amenability

Examples and counter-examples provided

Connection to property gamma and Kazhdan's T

Abstract

Let $G$ be a subgroup of a discrete (countable) group $\Gamma$. We introduce a notion of relative inner amenability of $G$ in $\Gamma$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss briefly the corresponding relative property gamma for pairs of finite factors and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property $T$.