Splittings of One-Ended Groups with One-Ended Halfspaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914082798436352 |
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| author | Mihalik, Michael Shepherd, Sam |
| author_facet | Mihalik, Michael Shepherd, Sam |
| contents | We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group $G$ admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then $H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected at infinity and $G$ is not an $n$-dimensional duality group for $n\geq3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06218 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Splittings of One-Ended Groups with One-Ended Halfspaces Mihalik, Michael Shepherd, Sam Group Theory 20F65 (Primary) 20F69, 20E08 (Secondary) We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group $G$ admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then $H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected at infinity and $G$ is not an $n$-dimensional duality group for $n\geq3$. |
| title | Splittings of One-Ended Groups with One-Ended Halfspaces |
| topic | Group Theory 20F65 (Primary) 20F69, 20E08 (Secondary) |
| url | https://arxiv.org/abs/2308.06218 |