Relative numbers of ends and quasi-median graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908947327221760 |
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| author | Genevois, Anthony |
| author_facet | Genevois, Anthony |
| contents | Given a finitely generated $G$ and a subgraph $H \leq G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{Sch}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite components of the complement of a neighbourhood of $H$ in $G$. Generalising Sageev's characterisation of codimension-one subgroups in terms of actions on CAT(0) cube complexes, we characterise the number of relative ends and the number of coends of a pair $(G,H)$ in terms of actions on quasi-median graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative numbers of ends and quasi-median graphs Genevois, Anthony Group Theory Combinatorics Geometric Topology Metric Geometry 20F65, 20F69 Given a finitely generated $G$ and a subgraph $H \leq G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{Sch}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite components of the complement of a neighbourhood of $H$ in $G$. Generalising Sageev's characterisation of codimension-one subgroups in terms of actions on CAT(0) cube complexes, we characterise the number of relative ends and the number of coends of a pair $(G,H)$ in terms of actions on quasi-median graphs. |
| title | Relative numbers of ends and quasi-median graphs |
| topic | Group Theory Combinatorics Geometric Topology Metric Geometry 20F65, 20F69 |
| url | https://arxiv.org/abs/2604.06686 |