Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions

We aim to interpret important constructions in the theory of automorphisms of the shift dynamical system in terms of subgroups $\mathcal{L}_{n,r}$ of the outer-automorphism groups $\mathcal{O}_{n,r}$ of the Higman--Thompson group $G_{n,r}$, and to extend results and techniques in $\operatorname{Aut}(X_n^{\mathbb{Z}}, \sigma_{n})$ to the groups of automorphisms $\operatorname{Aut}(G_{n,r})$ and outerautomrphisms of the Higman--Thompson group $G_{n,r}$. Our mains results are a concrete realisation of the "inert subgroup", important subgroup in the study of automorphism groups of shift spaces, as a subgroup $\mathcal{K}_{n}$ of $\mathcal{L}_{n,n-1}$. Using this realisation, we show that the $\operatorname{Aut}(G_{n,r})$ contains an isomorphic copy of $\operatorname{Aut}(X_{m}^{\mathbb{Z}}, \sigma_{m})$ for all $m \ge 2$. A survey of the literature then yields that $\operatorname{Aut}(G_{n,r})$ contains isomorphic copies of finite groups, finitely generated abelian groups, free groups, free products of finite groups, fundamental groups of 2-manifolds, graph groups and countable locally finite residually finite groups to name a few. We extend a result for $\operatorname{Aut}(X_n^{\mathbb{Z}}, \sigma_{n})$ to the group $\mathcal{O}_{n,n-1}$. The homeomorphism $\overleftarrow{\phantom{a}}$ of $X_n^{\mathbb{Z}}$ which maps a sequence $(x_i)_{i \in \mathbb{Z}}$ to the sequence $(y_{i})_{i \in \mathbb{Z}}$ defined such that $y_{i} = x_{-i}$ induces an automorphism $\overleftarrow{\mathfrak{r}}$ of $\operatorname{Aut}(X_n^{\mathbb{Z}}, \sigma_{n})$, and consequently, an automorphism of $\mathcal{L}_{n}$. We extend the automorphism $\overleftarrow{{\mathfrak{r}}}$ to the group $\mathcal{O}_{n,n-1}$. In a forthcoming article, we demonstrate that the group $\mathcal{O}_{n}$ is isomorphic to the mapping class group of the full two-sided shift over $n$ letters.