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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.24581 |
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| _version_ | 1866914118871547904 |
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| author | Boudec, Adrien Le |
| author_facet | Boudec, Adrien Le |
| contents | A locally compact group $G$ is a cocompact envelope of a group $Γ$ if $G$ contains a copy of $Γ$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $Γ,Λ$ having a common cocompact envelope, and asks what properties must be shared between $Γ$ and $Λ$.
We first consider the setting where the common cocompact envelope is totally disconnected. In that situation we show that if $Γ$ admits a finitely generated nilpotent normal subgroup $A$, then virtually $Λ$ admits a normal subgroup $B$ such that $A$ and $B$ are virtually isomorphic.
We establish both rigidity and flexibility results when $Γ$ belongs to the class of solvable groups of finite rank. On the rigidity perspective, we show that if $Γ$ is solvable of finite rank, and the locally finite radical of $Λ$ is finite, then $Λ$ must be virtually solvable of finite rank. On the flexibility perspective, we exhibit groups $Γ,Λ$ with a common cocompact envelope such that $Γ$ is solvable of finite rank, while $Λ$ is not virtually solvable. In particular the class of solvable groups of finite rank is not QI-rigid. We also exhibit flexibility behaviours among finitely presented groups, and more generally among groups with type $F_n$ for arbitrary $n \geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity and flexibility results for groups with a common cocompact envelope Boudec, Adrien Le Group Theory A locally compact group $G$ is a cocompact envelope of a group $Γ$ if $G$ contains a copy of $Γ$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $Γ,Λ$ having a common cocompact envelope, and asks what properties must be shared between $Γ$ and $Λ$. We first consider the setting where the common cocompact envelope is totally disconnected. In that situation we show that if $Γ$ admits a finitely generated nilpotent normal subgroup $A$, then virtually $Λ$ admits a normal subgroup $B$ such that $A$ and $B$ are virtually isomorphic. We establish both rigidity and flexibility results when $Γ$ belongs to the class of solvable groups of finite rank. On the rigidity perspective, we show that if $Γ$ is solvable of finite rank, and the locally finite radical of $Λ$ is finite, then $Λ$ must be virtually solvable of finite rank. On the flexibility perspective, we exhibit groups $Γ,Λ$ with a common cocompact envelope such that $Γ$ is solvable of finite rank, while $Λ$ is not virtually solvable. In particular the class of solvable groups of finite rank is not QI-rigid. We also exhibit flexibility behaviours among finitely presented groups, and more generally among groups with type $F_n$ for arbitrary $n \geq 1$. |
| title | Rigidity and flexibility results for groups with a common cocompact envelope |
| topic | Group Theory |
| url | https://arxiv.org/abs/2510.24581 |