Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges

Abstract

The recent breakthrough works [9, 11, 12] which established the amenability for new classes of groups, lead to the following question: is the action $W({\mathbb{Z}}^d)\curvearrowright {\mathbb{Z}}^d$ extensively amenable? (Where $W({\mathbb{Z}}^d)$ is the wobbling group of permutations $\sigma :{\mathbb{Z}}^d\to {\mathbb{Z}}^d$ with bounded range). This is equivalent to asking whether the action $(\mathbb{Z}/2\mathbb{Z}{)}^{({\mathbb{Z}}^d)}⋊W({\mathbb{Z}}^d)\curvearrowright (\mathbb{Z}/2\mathbb{Z}{)}^{({\mathbb{Z}}^d)}$ is amenable. The $d=1$ and $d=2$ and have been settled respectively in [9, 11]. By [12], a positive answer to this question would imply the amenability of the IET group. In this work, we give a partial answer to this question by introducing a natural strengthening of the notion of extensive-amenability which we call diffuse-extensive-amenability.

Our main result is that for any bounded degree graph $X$, the action $W(X)\curvearrowright X$ is diffuse-extensively amenable if and only if $X$ is recurrent. Our proof is based on the construction of suitable stochastic processes $({\tau }_t{)}_{t\ge 0}$ on $W(X)<\mathfrak{S}(X)$ whose inverted orbits ${\bar{O}}_t(x_0)=\{x\in X:\text{there exists }s\le t\text{ s.t. }{\tau }_s(x)=x_0\}={\bigcup }_{0\le s\le t}{\tau }_s^{-1}(\{x_0\})$ are exponentially unlikely to be sub-linear when $X$ is transient.

This result leads us to conjecture that the action $W({\mathbb{Z}}^d)\curvearrowright {\mathbb{Z}}^d$ is not extensively amenable when $d\ge 3$ and that a different route towards the (non-?)amenability of the IET group may be needed.