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

DOI10.4171/GGD/567arXiv1804.01981MaRDI QIDQ6299996

Publication date: 5 April 2018

Abstract: The recent breakthrough works [6,8,9] which established the amenability for new classes of groups, lead to the following question: is the action

W (m a t h b b Z^{d}) c u r v e a r r o w r i g h t m a t h b b Z^{d}

extensively amenable? (Where

W (m a t h b b Z^{d})

is the {em wobbling group} of permutations

s i g m a : m a t h b b Z^{d} o m a t h b b Z^{d}

with bounded range). This is equivalent to asking whether the action

(m a t h b b Z / 2 m a t h b b Z)^{(m a t h b b Z^{d})} t i m e s W (m a t h b b Z^{d}) c u r v e a r r o w r i g h t (m a t h b b Z / 2 m a t h b b Z)^{(m a t h b b Z^{d})}

is amenable. The

d = 1

and

d = 2

and have been settled respectively in [6,8]. By [9], 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) c u r v e a r r o w r i g h t X

is diffuse-extensively amenable if and only if

X

is recurrent. Our proof is based on the construction of suitable stochastic processes

(a u_{t})_{t g e q 0}

on

W (X), <, m a t h f r a k S (X)

whose {em inverted orbits}

�ar O_t(x_0) = {xin X, exists sleq t,, au_s(x)=x_0} = �igcup_{0leq s leq t} au_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 (m a t h b b Z^{d}) c u r v e a r r o w r i g h t m a t h b b Z^{d}

is not extensively amenable when

d g e q 3

and that a different route towards the (non-?)amenability of the IET group may be needed.

Mathematics Subject Classification ID

Extensions, wreath products, and other compositions of groups (20E22) Dynamics induced by group actions other than (mathbb{Z}) and (mathbb{R}), and (mathbb{C}) (37C85) Continuous-time Markov processes on discrete state spaces (60J27) Symmetric groups (20B30) Means on groups, semigroups, etc.; amenable groups (43A07) Subgroups of symmetric groups (20B35) Probabilistic methods in group theory (20P05) Applications of continuous-time Markov processes on discrete state spaces (60J28)

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