Local finiteness and automorphism groups of low complexity subshifts

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 $\frac{c_n(X)}{nf(n)} \rightarrow 0$.