On subgroups with narrow Schreier graphs

We study finitely generated pairs of groups $H\le G$ such that the Schreier graph of $H$ has at least two ends and is \emph{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.