Invariant means for the wobbling group

Abstract

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\to X$ satisfying $\underset{x\in X}{\sup }d(g(x),x)<\infty$. 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 ${⨁}_X\mathbf{Z}/2\mathbf{Z}⋊W(X)$ on ${⨁}_X\mathbf{Z}/2\mathbf{Z}$ and property (T).

8 pages. v3: final version, with new presentation; to appear in the Bulletin of the BMS