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 groups of automorphisms of shift spaces and the groups of outer automorphisms of the Higman-Thompson groups $\{G_{n,r}\}$. In previous work, the authors show that the group $\mathrm{Aut}(X_n^{\mathbb{N}}, \sigma_{n})$ of automorphisms of the one-sided shift dynamical system over an $n$-letter alphabet naturally embeds as a subgroup of the group $\mathop{\mathrm{Out}}(G_{n,r})$ of outer-automorphisms of the Higman-Thompson group $G_{n, r}$, $1 \le r < n$. In the current article we show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathop{\mathrm{Aut}}(X_n^{\mathbb{Z}}, \sigma_{n})$ by its centre embeds as a subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,r})$ of $G_{n,r}$. It follows by a result of Ryan that we have the following central extension: $$1 \to \langle \sigma_{n}\rangle \to \mathrm{Aut}(X_n^{\mathbb{Z}}, \sigma_{n}) \to \mathcal{L}_{n}.$$ A consequence of this is that the groups $\mathrm{Out}(G_{n,r})$ are centreless and have undecidable order problem.