#### 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 c_{n}(X), if lim inf(log(c_{n}(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 c_{n}(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 c_{n}(X)/n^{2}(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 c_{n}(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

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

#### Recommended Citation

Pavlov, Ronnie and Schmieding, Scott, "Local Finiteness and Automorphism Groups of Low Complexity Subshifts" (2022). *Mathematics: Faculty Scholarship*. 56.

https://digitalcommons.du.edu/math_faculty/56

#### Copyright Statement / License for Reuse

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