The recent breakthrough works [9, 11, 12] which established the amenability for new classes of groups, lead to the following question: is the action extensively amenable? (Where is the wobbling group of permutations with bounded range). This is equivalent to asking whether the action is amenable. The and and have been settled respectively in [9, 11]. By , 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 , the action is diffuse-extensively amenable if and only if is recurrent. Our proof is based on the construction of suitable stochastic processes on whose inverted orbits
are exponentially unlikely to be sub-linear when is transient.
This result leads us to conjecture that the action is not extensively amenable when 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. 871–897DOI 10.4171/GGD/567