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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.19000 |
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| _version_ | 1866910576018456576 |
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| author | Azuelos, Pénélope |
| author_facet | Azuelos, Pénélope |
| contents | We study finitely generated pairs of groups $H \leq G$ such that the Schreier graph of $H$ has at least two ends and is \emph{narrow}. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that $H$ is a virtual fiber subgroup if and only if $G$ contains infinitely many double cosets of $H$. Along the way, we prove that if a group acts essentially on a finite dimensional CAT(0) cube complex with no facing triples then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On subgroups with narrow Schreier graphs Azuelos, Pénélope Group Theory 20F65 We study finitely generated pairs of groups $H \leq G$ such that the Schreier graph of $H$ has at least two ends and is \emph{narrow}. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that $H$ is a virtual fiber subgroup if and only if $G$ contains infinitely many double cosets of $H$. Along the way, we prove that if a group acts essentially on a finite dimensional CAT(0) cube complex with no facing triples then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser. |
| title | On subgroups with narrow Schreier graphs |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2402.19000 |