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

### Christophe Garban

Université Claude Bernard Lyon 1, Villeurbanne, France

## 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*

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.

## Cite this article

Christophe Garban, Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges. Groups Geom. Dyn. 14 (2020), no. 3, pp. 871–897

DOI 10.4171/GGD/567