  Suppose $(X,\sigma)$ is a subshift, $P_X(n)$ is the word complexity function
of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that
if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable,
discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$
can never contain a nonabelian free semigroup (and, in particular, can never
contain a nonabelian free subgroup). This is in contrast to recent examples,
due to Salo and Schraudner, of subshifts with quadratic complexity that do
contain such a semigroup.
