Amenability of linear-activity automaton groups

Abstract

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.

29 pages, 1 figure. Revised version after referee report. To appear in the Journal of the European Mathematical Society

References in corpus (2)

• Amenability via random walks
• Positive speed for high-degree automaton groups

Cited by in corpus (12)

• Extensions of amenable groups by recurrent groupoids
• Varieties
• Positive speed for high-degree automaton groups
• Extensive amenability and an application to interval exchanges
• Behaviors of entropy on finitely generated groups
• The Liouville property for groups acting on rooted trees
• Schreier graphs of spinal groups
• Speed exponents of random walks on groups
• (Self-)similar groups and the Farrell-Jones conjectures
• Growth of Schreier graphs of automaton groups
• Amenability of Bounded Automata Groups on Infinite Alphabets
• Subshifts with slow complexity and simple groups with the Liouville property