The mapping class group of a shift of finite type
Mike Boyle, Sompong Chuysurichay · Journal of Modern Dynamics · 2018
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.