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 $\sigma\colon \mathbb Z^d \to \mathbb Z^d$ with bounded range). This is equivalent to asking whether the action $(\mathbb Z/2\mathbb Z)^{(\mathbb Z^d)} \rtimes W(\mathbb Z^d) \curvearrowright (\mathbb Z/2\mathbb Z)^{(\mathbb Z^d)}$ 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 $(\tau\_t)\_{t\geq 0}$ on $W(X), <, \mathfrak{S}(X)$ whose inverted orbits $$ \bar O\_t(x\_0) = {x\in X\colon \text{there exists } s\leq t \text{\ s.t.\ } \tau\_s(x)=x\_0} = \bigcup\_{0\leq s \leq t} \tau\_s^{-1}({x\_0}) $$ 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 $d\geq 3$ and that a different route towards the (non-?)amenability of the IET group may be needed.