Local finiteness and automorphism groups of low complexity subshifts

• Published in: Ergodic theory and dynamical systems Vol. 43; no. 6; pp. 1980 - 2001
• Main Authors: PAVLOV, RONNIE, SCHMIEDING, SCOTT
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2023
• Subjects: Automorphisms; Complexity; Group theory; Hypotheses; Original Article; Subgroups; Upper bounds; 20F65; word complexity; symbolic dynamics; automorphism groups; 37B10; locally finite groups
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2022.7

We prove that for any transitive subshift X with word complexity function $c_n(X)$ , if $\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$ , then the quotient group ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ 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 ${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$ , then $\mathrm {Aut}(X,\sigma )$ 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 ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ isomorphic to G and ${c_n(X)}/{nf(n)} \rightarrow 0$ .