The automorphism group of a shift of slow growth is amenable
Van Cyr, Bryna Kra · Ergodic Theory and Dynamical Systems · 2018
Suppose $(X,\unicode[STIX]{x1D70E})$ is a subshift, $P_{X}(n)$ is the word complexity function of $X$ , and $\text{Aut}(X)$ is the group of automorphisms of $X$ . We show that if $P_{X}(n)=o(n^{2}/\log ^{2}n)$ , then $\text{Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_{X}(n)=o(n^{2})$ , then $\text{Aut}(X)$ can never contain a non-abelian free monoid (and, in particular, can never contain a non-abelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a monoid.