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The automorphism group of a shift of slow growth is amenable
Published online by Cambridge University Press: 11 December 2018
Abstract
Suppose $(X,\unicode[STIX]{x1D70E})$ is a subshift,
$P_{X}(n)$ is the word complexity function of
$X$, and
$\text{Aut}(X)$ is the group of automorphisms of
$X$. We show that if
$P_{X}(n)=o(n^{2}/\log ^{2}n)$, then
$\text{Aut}(X)$ is amenable (as a countable, discrete group). We further show that if
$P_{X}(n)=o(n^{2})$, then
$\text{Aut}(X)$ can never contain a non-abelian free monoid (and, in particular, can never contain a non-abelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a monoid.
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- Original Article
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- © Cambridge University Press, 2018
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