Groups, Geometry, and Dynamics

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Volume 8, Issue 1, 2014, pp. 23–38
DOI: 10.4171/GGD/215

Published online: 2014-05-13

Positive speed for high-degree automaton groups

Gideon Amir[1] and Bálint Virág[2]

(1) Bar-Ilan University, Ramat Gan, Israel
(2) University of Toronto, Toronto, Canada

Mother groups are the basic building blocks for polynomial automaton groups. We show that, in contrast with mother groups of degree 0 or 1, any bounded, symmetric, generating random walk on the mother groups of degree at least 3 has positive speed. The proof is based on an analysis of resistance in fractal mother graphs. We give upper bounds on resistances in these graphs, and show that infinite versions are transient.

Keywords: Automaton groups, random walks, Liouville property

Amir Gideon, Virág Bálint: Positive speed for high-degree automaton groups. Groups Geom. Dyn. 8 (2014), 23-38. doi: 10.4171/GGD/215