Positive speed for high-degree automaton groups

Abstract

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 tran- sient.

18 pages, 4 figures

References in corpus (2)

• Amenability via random walks
• Amenability of linear-activity automaton groups

Cited by in corpus (9)

• Amenability of linear-activity automaton groups
• Extensions of amenable groups by recurrent groupoids
• Extensive amenability and an application to interval exchanges
• Behaviors of entropy on finitely generated groups
• The Liouville property for groups acting on rooted trees
• Speed exponents of random walks on groups
• Non-Liouville groups with return probability exponent at most 1/2
• Random walks on the discrete affine group
• Isoperimetric profiles and random walks on some groups defined by piecewise actions