TY - JOUR

T1 - Local finiteness and automorphism groups of low complexity subshifts

AU - Pavlov, Ronnie

AU - Schmieding, Scott

N1 - Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press.

PY - 2023

Y1 - 2023

N2 - We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/log log log n))=0, then the quotient group Aut(X, σ)/σ) of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(X) 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 cn(X)/n2(logn)-1 → 0, then Aut (X, σ) 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 f: N → N, there exists a minimal subshift X with Aut(X,σ)/(σ) isomorphic to G and cn(X)/nf(n)→ 0.

AB - We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/log log log n))=0, then the quotient group Aut(X, σ)/σ) of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(X) 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 cn(X)/n2(logn)-1 → 0, then Aut (X, σ) 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 f: N → N, there exists a minimal subshift X with Aut(X,σ)/(σ) isomorphic to G and cn(X)/nf(n)→ 0.

UR - http://www.scopus.com/inward/record.url?scp=85129287339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85129287339&partnerID=8YFLogxK

U2 - 10.1017/etds.2022.7

DO - 10.1017/etds.2022.7

M3 - Article

AN - SCOPUS:85129287339

SN - 0143-3857

VL - 43

SP - 1980

EP - 2001

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 6

ER -