Automorphism groups of solvable groups of finite abelian ranks
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912918674604032 |
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| author | Deré, Jonas Pengitore, Mark |
| author_facet | Deré, Jonas Pengitore, Mark |
| contents | This paper gives a new explicit construction of the $\mathbb{Q}$-algebraic hull for virtually solvable groups $Γ$ of finite abelian ranks, taking into account the spectrum $S$ of the group $Γ$. As an application, we make a detailed study of the structure of $Aut(Γ)$ in the finitely generated case and show that a number of natural subgroups are $S$-arithmetic under the condition that $Fitt(Γ)$ is $S$-arithmetic. We then proceed by demonstrating that $Out(Γ)$ has a $S$-arithmetic image in the group of algebraic outer automorphisms of the $\mathbb{Q}$-algebraic hull. We finish by discussing further applications of the $\mathbb{Q}$-algebraic hull towards an open conjecture by Nekrashevych and Pete and topological fixed point theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphism groups of solvable groups of finite abelian ranks Deré, Jonas Pengitore, Mark Group Theory This paper gives a new explicit construction of the $\mathbb{Q}$-algebraic hull for virtually solvable groups $Γ$ of finite abelian ranks, taking into account the spectrum $S$ of the group $Γ$. As an application, we make a detailed study of the structure of $Aut(Γ)$ in the finitely generated case and show that a number of natural subgroups are $S$-arithmetic under the condition that $Fitt(Γ)$ is $S$-arithmetic. We then proceed by demonstrating that $Out(Γ)$ has a $S$-arithmetic image in the group of algebraic outer automorphisms of the $\mathbb{Q}$-algebraic hull. We finish by discussing further applications of the $\mathbb{Q}$-algebraic hull towards an open conjecture by Nekrashevych and Pete and topological fixed point theory. |
| title | Automorphism groups of solvable groups of finite abelian ranks |
| topic | Group Theory |
| url | https://arxiv.org/abs/2506.07991 |