Relative inner amenability, relative property gamma and non-Kazhdan groups

TL;DR

This paper introduces the concept of relative inner amenability for subgroups within discrete groups, explores its properties, and connects it to the relative property gamma, leading to a characterization of non-Kazhdan groups.

Contribution

It defines and analyzes relative inner amenability and property gamma, providing new criteria and examples for understanding non-Kazhdan groups.

Findings

Established equivalent conditions for relative inner amenability.

Provided examples and counter-examples illustrating the concepts.

Characterized discrete, icc groups without Kazhdan's property (T).

Abstract

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