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