Document Type

Article

Publication Date

2022

Keywords

Symbolic dynamics, Word complexity, Automorphism groups, Locally finite groups

Organizational Units

College of Natural Science and Mathematics, Mathematics

Abstract

We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if cn(X)/n2(log n) −1 → 0, then Aut(X, σ) is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing f : N → N, there exists a minimal subshift X with Aut(X, σ)/〈 σ〉 isomorphic to G and cn(X)/nf (n) → 0.

Publication Statement

This article was originally published as:

Pavlov, R., & Schmieding, S. (2022). Local finiteness and automorphism groups of low complexity subshifts. Ergodic Theory and Dynamical Systems, 1-22. doi:10.1017/etds.2022.7

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Copyright Statement / License for Reuse

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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