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arXiv · 2604.06686

Relative numbers of ends and quasi-median graphs

Anthony Genevois
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Abstract

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.

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Anthony Genevois. 2026-04-08. Relative numbers of ends and quasi-median graphs. https://arxiv.org/abs/2604.06686

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