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 $\mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n})$ by its centre embeds as a particular subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$\langle \sigma_{n}\rangle \hookrightarrow \mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n}) \twoheadrightarrow \mathcal{L}_{n}$$ 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 $\mathop{\mathrm{Out}}(G_{n,r})$ are centreless and have undecidable order problem. Note that the group $\mathop{\mathrm{Out}}(G_{n,n-1})$ consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group $\mathcal{L}_{n}$ are easily characterised within $\mathop{\mathrm{Out}}(G_{n,n-1})$ 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 $\mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n})$, and we demonstrate how to compute products using this new representation.