On subgroups with narrow Schreier graphs

We study finitely generated pairs of groups such that the Schreier graph of has at least two ends and is 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 is a virtual fiber subgroup if and only if contains infinitely many double cosets of . 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.