Local finiteness and automorphism groups of low complexity subshifts

TL;DR

This paper investigates the relationship between the complexity of subshifts and the structure of their automorphism groups, establishing conditions under which these groups are locally finite or amenable, and constructing examples with prescribed automorphism groups.

Contribution

It provides new bounds linking word complexity to automorphism group properties, proves local finiteness under specific growth conditions, and constructs minimal subshifts with arbitrary countable automorphism groups.

Findings

Automorphism groups are locally finite under certain low complexity conditions.

Subshifts with complexity growing slower than quadratic imply amenability of automorphism groups.

Constructed minimal subshifts with prescribed countable automorphism groups.

Abstract

We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,\sigma) / \langle \sigma\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $\sigma$ 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 $\frac{c_n(X)}{n^2 (\log n)^{-1}} \rightarrow 0$, then $\textrm{Aut}(X,\sigma)$ 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 $f: \mathbb{N} \rightarrow \mathbb{N}$, there exists a minimal subshift $X$ with $\textrm{Aut}(X,\sigma) / \langle \sigma\rangle$ isomorphic to $G$ and $\frac{c_n(X)}{nf(n)} \rightarrow 0$.