Local finiteness and automorphism groups of low complexity subshifts

We prove that for any transitive subshift X with word complexity function, if, then the quotient group of the automorphism group of X by the subgroup generated by the shift is locally finite. We prove that significantly weaker upper bounds on 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, then 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, there exists a minimal subshift X with isomorphic to G and.