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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.18720 |
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| _version_ | 1866917734544048128 |
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| author | Olukoya, Feyishayo |
| author_facet | Olukoya, Feyishayo |
| contents | 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ 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 $\operatorname{Aut}(X_{m}^{\mathbb{Z}}, σ_{m})$ 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18720 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions Olukoya, Feyishayo Group Theory 20E36, 20F10, 37B10, 54H15 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ 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 $\operatorname{Aut}(X_{m}^{\mathbb{Z}}, σ_{m})$ 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$ 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 $\operatorname{Aut}(X_n^{\mathbb{Z}}, σ_{n})$, 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. |
| title | Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions |
| topic | Group Theory 20E36, 20F10, 37B10, 54H15 |
| url | https://arxiv.org/abs/2407.18720 |