Invariant means for the wobbling group

TL;DR

This paper investigates the algebraic and analytic properties of the wobbling group of a metric space, exploring its connections with amenability and property (T) in relation to the space's structure.

Contribution

It introduces new insights into the properties of the wobbling group and its actions, linking geometric features of the space with group-theoretic and analytic properties.

Findings

Characterization of amenability of the lamplighter group action

Analysis of property (T) for wobbling groups

Connections between metric space geometry and group properties

Abstract

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x)<\infty$. We study algebraic and analytic properties of $W(X)$ in relation with the metric space structure of $X$, such as amenability of the action of the lamplighter group $ \bigoplus_{X} \mathbf Z/2\mathbf Z \rtimes W(X)$ on $\bigoplus_{X} \mathbf Z/2\mathbf Z$ and property (T).