The mapping class group of a shift of finite type

Abstract

Copyright © 2017, arXiv, All rights reserved. Let (XA, σA) be a nontrivial irreducible shift of finite type (SFT), with MA denoting its mapping class group: the group of flow equivalences of its mapping torus SXA, (i.e., self homeomorphisms of SXA which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of SXA isotopic to the identity. We develop and apply machinery (flow codes, cohomology constraints) and provide context for the study of MA, and prove results including the following. MA acts faithfully and n-transitively (for every n in N) by permutations on the set of circles of SXA. The center of MA is trivial. The outer automorphism group of MA is nontrivial. In many cases, Aut(σA) admits a nonspatial automorphism. For every SFT (XB, σB) flow equivalent to (XA, σA), MA contains embedded copies of Aut(σB)/ hσBi, induced by return maps to invariant cross sections; but, elements of MA not arising from flow equivalences with invariant cross sections are abundant. MA is countable and has solvable word problem. MA is not residually finite. Conjugacy classes of many (possibly all) involutions in MA can be classified by the G-flow equivalence classes of associated G-SFTs, for G = Z/2Z. There are many open questions.