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  1. ; (, Ergodic Theory and Dynamical Systems)
    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. 
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