# 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 $W(Z_{d})↷Z_{d}$ extensively amenable? (Where $W(Z_{d})$ is the *wobbling group* of permutations $σ:Z_{d}→Z_{d}$ with bounded range). This is equivalent to asking whether the action $(Z/2Z)_{(Z_{d})}⋊W(Z_{d})↷(Z/2Z)_{(Z_{d})}$ 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 $W(X)↷X$ is diffuse-extensively amenable if and only if $X$ is recurrent. Our proof is based on the construction of suitable stochastic processes $(τ_{t})_{t≥0}$ on $W(X)<S(X)$ whose *inverted orbits* $Oˉ_{t}(x_{0})={x∈X:there existss≤ts.t.τ_{s}(x)=x_{0}}=⋃_{0≤s≤t}τ_{s}({x_{0}})$ are exponentially unlikely to be sub-linear when $X$ is transient.

This result leads us to conjecture that the action $W(Z_{d})↷Z_{d}$ is not extensively amenable when $d≥3$ 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