We prove that for any transitive subshift X with word complexity function
$c_n(X)$
, if
$\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$
, then the quotient group
${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$
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
${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$
, then
$\mathrm {Aut}(X,\sigma )$
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
${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$
isomorphic to G and
${c_n(X)}/{nf(n)} \rightarrow 0$
.