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Groups, Geometry, and Dynamics

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Volume 14, Issue 3, 2020, pp. 871–897
DOI: 10.4171/GGD/567

Published online: 2020-10-13

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

Christophe Garban[1]

(1) Université Claude Bernard Lyon 1, Villeurbanne, France

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.

Keywords: Amenable groups, extensive amenability, inverted orbits, exclusion process, group of Interval Exchanges Transformations

Garban Christophe: Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges. Groups Geom. Dyn. 14 (2020), 871-897. doi: 10.4171/GGD/567