Amenability of linear-activity automaton groups

By Gideon Amir, Omer Angel, B'alint Vir'ag. 2009

59 citations recorded by the source corpus.

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a sufficiently symmetric random walk on a specially constructed degree 1 automaton group — the mother group — has asymptotic entropy 0. Our result answers an open question by Nekrashevich in the Kourovka notebook, and gives a partial answer to a question of Sidki.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b093358079cd5095f00563af11efd6296ce05bd

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Citations in the corpus

Milnor's Problem on the Growth of Groups and its Consequences (2011, methodology): “It received further development in the paper of V. Kaimanovich [Kai05], where it was called the Munchhausen trick, and in the papers of Bartholdi, Kaimanovich and Nekrashevych [BKN10], and of Amir, Angel and Virag [AAV09].”
Extensions of amenable groups by recurrent groupoids (2013, methodology): “ocally ﬁnite. For example, one can use Theorem 1.1 twice: prove amenability of [[H]] using it, and then construct new amenable groups G using H. This way one gets a simple proof of the main result of [AAV13]: that groups generated by ﬁnite automata of linear activity growth are amenable. Two other e”
Extensive amenability and an application to interval exchanges (2015, background): “] for details). Amenability of polynomial activity automata groups is an open problem, that has been answered aﬃrmatively for d ≤ 2 [JNdlS13]. Concerning the Liouville property, it was conjectured in [AAV13] that it holds up to d = 2; this is known for d = 0 [BKN10, AAMBV13] and for d = 1 in some im”