Local finiteness and automorphism groups of low complexity subshifts

We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/log log log n))=0, then the quotient group Aut(X, σ)/σ) of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(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 cn(X)/n2(logn)-1 → 0, then Aut (X, σ) 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: N → N, there exists a minimal subshift X with Aut(X,σ)/(σ) isomorphic to G and cn(X)/nf(n)→ 0.