Description:
<jats:title>Abstract</jats:title><jats:p>We prove that for any transitive subshift <jats:italic>X</jats:italic> with word complexity function <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline1.png" /><jats:tex-math>
$c_n(X)$
</jats:tex-math></jats:alternatives></jats:inline-formula>, if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline2.png" /><jats:tex-math>
$\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$
</jats:tex-math></jats:alternatives></jats:inline-formula>, then the quotient group <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline3.png" /><jats:tex-math>
${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$
</jats:tex-math></jats:alternatives></jats:inline-formula> of the automorphism group of <jats:italic>X</jats:italic> by the subgroup generated by the shift <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline4.png" /><jats:tex-math>
$\sigma $
</jats:tex-math></jats:alternatives></jats:inline-formula> is locally finite. We prove that significantly weaker upper bounds on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline5.png" /><jats:tex-math>
$c_n(X)$
</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:italic>X</jats:italic> of range <jats:italic>n</jats:italic> in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift <jats:italic>X</jats:italic>, if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline6.png" /><jats:tex-math>
${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$
</jats:tex-math></jats:alternatives></jats:inline-formula>, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline7.png" /><jats:tex-math>
$\mathrm {Aut}(X,\sigma )$
</jats:tex-math></jats:alternatives></jats:inline-formula> is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group <jats:italic>G</jats:italic> and any unbounded increasing <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline8.png" /><jats:tex-math>
$f: \mathbb {N} \rightarrow \mathbb {N}$
</jats:tex-math></jats:alternatives></jats:inline-formula>, there exists a minimal subshift <jats:italic>X</jats:italic> with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline9.png" /><jats:tex-math>
${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$
</jats:tex-math></jats:alternatives></jats:inline-formula> isomorphic to <jats:italic>G</jats:italic> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000074_inline10.png" /><jats:tex-math>
${c_n(X)}/{nf(n)} \rightarrow 0$
</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>