# CAT(0) cube complexes and inner amenability

## TL;DR

This paper investigates inner amenability in groups acting on CAT(0) cube complexes, providing geometric characterizations and extending results to trees, wreath products, and using novel methods involving conjugation-invariant means.

## Contribution

It introduces new geometric and algebraic characterizations of inner amenability for groups acting on CAT(0) cube complexes, including wreath products and amalgamated free products.

## Key findings

- Characterization of inner amenability for groups acting on CAT(0) cube complexes
- Complete characterization of inner amenability for wreath products
- Introduction of the location lemma for conjugation-invariant means

## Abstract

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on $G$. We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group $G$ relative to a given normal subgroup $N$ of $G$. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.