On commensurators of free groups and free pro-p groups
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918137150046208 |
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| author | Barnea, Yiftach Ershov, Mikhail Boudec, Adrien Le Reid, Colin D. Vannacci, Matteo Weigel, Thomas |
| author_facet | Barnea, Yiftach Ershov, Mikhail Boudec, Adrien Le Reid, Colin D. Vannacci, Matteo Weigel, Thomas |
| contents | We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple.
For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency.
If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On commensurators of free groups and free pro-p groups Barnea, Yiftach Ershov, Mikhail Boudec, Adrien Le Reid, Colin D. Vannacci, Matteo Weigel, Thomas Group Theory Primary: 20E05, 20E32, Secondary 20E18, 20F28, 22D05 We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple. |
| title | On commensurators of free groups and free pro-p groups |
| topic | Group Theory Primary: 20E05, 20E32, Secondary 20E18, 20F28, 22D05 |
| url | https://arxiv.org/abs/2507.04120 |