Publication Date

2022

Document Type

Article

Organizational Units

College of Natural Science and Mathematics, Mathematics

Keywords

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

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.

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.

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

Copyright is held by the authors. User is responsible for all copyright compliance.

Rights Holder

Ronnie Pavlov, Scott Schmieding

Provenance

Received from author

File Format

application/pdf

Language

English (eng)

Extent

22 pgs

File Size

400 KB

Publication Title

Ergodic Theory and Dynamical Systems

Volume

43

First Page

1

Last Page

22

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