Property (LR) and an embedding theorem for virtually free groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915877390123008 |
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| author | Minasyan, Ashot |
| author_facet | Minasyan, Ashot |
| contents | We prove that every virtually free group $G$ has property (LR) of Long and Reid: each finitely generated subgroup of $G$ is a retract of a finite index subgroup. The main ingredient in the proof is a new embedding result stating that every countable virtually free group embeds in a double of a finite group.
As a corollary, we show that any group commensurable with the direct product of a free group and a finitely generated abelian group has (LR). This applies to generalized Baumslag-Solitar groups of arbitrary rank $n \in \mathbb{N}$ with finite monodromy, which, in particular, include all non-cyclic one-relator groups with center. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17596 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Property (LR) and an embedding theorem for virtually free groups Minasyan, Ashot Group Theory 20E05, 20E25, 20E06, 20E07 We prove that every virtually free group $G$ has property (LR) of Long and Reid: each finitely generated subgroup of $G$ is a retract of a finite index subgroup. The main ingredient in the proof is a new embedding result stating that every countable virtually free group embeds in a double of a finite group. As a corollary, we show that any group commensurable with the direct product of a free group and a finitely generated abelian group has (LR). This applies to generalized Baumslag-Solitar groups of arbitrary rank $n \in \mathbb{N}$ with finite monodromy, which, in particular, include all non-cyclic one-relator groups with center. |
| title | Property (LR) and an embedding theorem for virtually free groups |
| topic | Group Theory 20E05, 20E25, 20E06, 20E07 |
| url | https://arxiv.org/abs/2603.17596 |