Distortion and the automorphism group of a shift

Van Cyr; John Franks; Bryna Kra; Samuel Petite

doi:10.3934/jmd.2018015

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2018, Volume 13: 147-161. Doi: 10.3934/jmd.2018015

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Distortion and the automorphism group of a shift

1.
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA
2.
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
3.
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France

Dedicated to the memory of Roy Adler

Received: December 15, 2016

Revised: February 22, 2017

Published: December 2018

BK: Partially supported by NSF grant 1500670.

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Abstract

The set of automorphisms of a one-dimensional subshift $(X, σ)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy subshift. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm top}}(X) = 0$ . We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 22F50, 37B15, 54H20.

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