On subgroups with narrow Schreier graphs

We study finitely generated pairs of groups H G such that the Schreier graph of H 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 H is a virtual fiber subgroup if and only if G contains infinitely many double cosets of H. 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.