Automorphisms of shift spaces and the Higman-Thompson groups: the two-sided case

In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman–Thompson groups $\{G_{n,r}\}$. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathrm{A}\mathrm{u}\mathrm{t}(X_n^{\mathbb{N}},{\sigma }_n)$ by its centre embeds as a particular subgroup ${\mathcal{L}}_n$ of the outer automorphism group $(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$⟨{\sigma }_n⟩\hookrightarrow \mathrm{A}\mathrm{u}\mathrm{t}(X_n^{\mathbb{N}},{\sigma }_n)\twoheadrightarrow {\mathcal{L}}_n$$ where here, $\langle \sigma_{n} \rangle \cong $\mathbb{Z}$$. We prove that this short exact sequence splits if and only if $n$ is not a proper power, and, in all cases, we compute the 2-cocycles and 2-coboundaries for the extension. We also use this central extension to prove that for $1\le r<n$, the groups $(G_{n,r})$ are centreless and have undecidable order problem. Note that the group $(G_{n,n-1})$ consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group ${\mathcal{L}}_n$ are easily characterised within $(G_{n,n-1})$ by a simple combinatorial property. In particular, the short exact sequence allows us to determine a new and efficient purely combinatorial representation of elements of $\mathrm{A}\mathrm{u}\mathrm{t}(X_n^{\mathbb{N}},{\sigma }_n)$, and we demonstrate how to compute products using this new representation.