TY - JOUR

T1 - Local finiteness and automorphism groups of low complexity subshifts

AU - Pavlov, Ronnie

AU - Schmieding, S. C.O.T.T.

N1 - Funding Information:
R.P. gratefully acknowledges the support of a Simons Foundation Collaboration Grant.
Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press.

PY - 2022

Y1 - 2022

N2 - We prove that for any transitive subshift X with word complexity function, if, then the quotient group of the automorphism group of X by the subgroup generated by the shift is locally finite. We prove that significantly weaker upper bounds on 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 X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if, then is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing, there exists a minimal subshift X with isomorphic to G and.

AB - We prove that for any transitive subshift X with word complexity function, if, then the quotient group of the automorphism group of X by the subgroup generated by the shift is locally finite. We prove that significantly weaker upper bounds on 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 X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if, then is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing, there exists a minimal subshift X with isomorphic to G and.

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U2 - 10.1017/etds.2022.7

DO - 10.1017/etds.2022.7

M3 - Article

AN - SCOPUS:85129287339

SN - 0143-3857

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -