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The automorphism group of a shift of slow growth is amenable

Part of: Discrete mathematics in relation to computer science Abstract harmonic analysis Topological dynamics

Published online by Cambridge University Press:  11 December 2018

VAN CYR  and
BRYNA KRA
VAN CYR
Affiliation:
Bucknell University, Lewisburg, PA 17837, USA email van.cyr@bucknell.edu
BRYNA KRA
Affiliation:
Northwestern University, Evanston, IL 60208, USA email kra@math.northwestern.edu
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Abstract

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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.

Keywords

subshiftautomorphismblock complexityamenable

MSC classification

Primary: 37B50: Multi-dimensional shifts of finite type, tiling dynamics
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1788 - 1804
Copyright
© Cambridge University Press, 2018

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