We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.
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Gideon Amir, Omer Angel, Bálint Virág, Amenability of linear-activity automaton groups. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 705–730DOI 10.4171/JEMS/373