Relative numbers of ends and quasi-median graphs

TL;DR

This paper characterizes the relative number of ends and coends of a subgroup within a finitely generated group using actions on quasi-median graphs, generalizing Sageev's work on CAT(0) cube complexes.

Contribution

It introduces a new framework to understand subgroup ends and coends via quasi-median graphs, extending Sageev's characterization to a broader class of graphs.

Findings

Characterizes relative ends and coends using quasi-median graphs.

Generalizes Sageev's characterization from CAT(0) cube complexes.

Provides a new perspective on subgroup structure in finitely generated groups.

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.