The mapping class group of a shift of finite type

Mike Boyle; Sompong Chuysurichay

doi:10.3934/jmd.2018014

Article Contents

2018, Volume 13: 115-145. Doi: 10.3934/jmd.2018014

This volume Previous Article Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions Next Article Periodic Reeb orbits on prequantization bundles

The mapping class group of a shift of finite type

Mike Boyle^1, and
Sompong Chuysurichay^2,

1.
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA
2.
Algebra and Applications Research Unit, Department of Mathematics and Statistics, Prince of Songkla University, Songkhla, Thailand 90110

Dedicated to Roy Adler, in memory of his insight, humor, and kindness

Received: April 27, 2017

Revised: August 18, 2017

Published: December 2018

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Abstract

Let $(X_A,σ_{A})$ be a nontrivial irreducible shift of finite type (SFT), with $\mathscr{M}_A$ denoting its mapping class group: the group of flow equivalences of its mapping torus $\mathsf{S} X_A$, (i.e., self homeomorphisms of $\mathsf{S} X_A$ which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of $\mathsf{S} X_A$ isotopic to the identity. We develop and apply machinery (flow codes, cohomology constraints) and provide context for the study of $\mathscr M_A$, and prove results including the following. $\mathscr{M}_A$ acts faithfully and $n$-transitively (for every $n$ in $\mathbb{N}$) by permutations on the set of circles of $\mathsf{S} X_A$. The center of $\mathscr{M}_A$ is trivial. The outer automorphism group of $\mathscr{M}_A$ is nontrivial. In many cases, $\text{Aut}(σ_{A})$ admits a nonspatial automorphism. For every SFT $(X_B,σ_B)$ flow equivalent to $(X_A,σ_{A})$, $\mathscr{M}_A$ contains embedded copies of ${\rm{Aut}}({\sigma _B})/\left\langle {{\sigma _B}} \right\rangle $, induced by return maps to invariant cross sections; but, elements of $\mathscr M_A$ not arising from flow equivalences with invariant cross sections are abundant. $\mathscr{M}_A$ is countable and has solvable word problem. $\mathscr{M}_A$ is not residually finite. Conjugacy classes of many (possibly all) involutions in $\mathscr M_A$ can be classified by the $G$-flow equivalence classes of associated $G$-SFTs, for $G = \mathbb{Z}/2\mathbb{Z}$. There are many open questions.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 20F10, 20F38.

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