Inverted Orbits Of Exclusion Processes, Diffuse-Extensive-Amenability, And (Non-?)Amenability Of The Interval Exchanges

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)) curved right arrow Z(d) extensively amenable? (Where W(Z(d)) is the wobbling group of permutations sigma: Z(d) -> Z(d) with bounded range). This is equivalent to asking whether the action (Z/2Z)((Zd)) (sic) W(Z(d)) curved right arrow (Z/2Z)((Zd)) 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) curved right arrow 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 >= 0) on W(X) < S(X) whose inverted orbits<(O)over bar>(t)(x(0)) = {x is an element of X : there exists s <= t s.t. tau(s)(x) = x(0)} = boolean OR(0 <= s <= t) tau(-1)(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)) curved right arrow 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.