JournalsggdVol. 14, No. 3pp. 871–897

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

  • Christophe Garban

    Université Claude Bernard Lyon 1, Villeurbanne, France
Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges cover
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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(Zd)ZdW(\mathbb Z^d) \curvearrowright \mathbb Z^d extensively amenable? (Where W(Zd)W(\mathbb Z^d) is the wobbling group of permutations σ ⁣:ZdZd\sigma\colon \mathbb Z^d \to \mathbb Z^d with bounded range). This is equivalent to asking whether the action (Z/2Z)(Zd)W(Zd)(Z/2Z)(Zd)(\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=1d = 1 and d=2d = 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 XX, the action W(X)XW(X)\curvearrowright X is diffuse-extensively amenable if and only if XX is recurrent. Our proof is based on the construction of suitable stochastic processes (τt)t0(\tau_t)_{t\geq 0} on W(X)<S(X)W(X)\, <\, \mathfrak{S}(X) whose inverted orbits

Oˉt(x0)={xX ⁣:there exists st s.t. τs(x)=x0}=0stτs1({x0})\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 XX is transient.

This result leads us to conjecture that the action W(Zd)ZdW(\mathbb Z^d)\curvearrowright \mathbb Z^d is not extensively amenable when d3d\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