Quasi-isometries between graphs of totally disconnected locally compact groups

Abstract

Let $G$ and $H$ be compactly generated totally disconnected locally compact (tdlc) groups that decompose as finite graphs of tdlc groups $(\mathcal{G}, \mathcal{A})$ and $(\mathcal{H}, \mathcal{B})$ such that all edge groups are compact and all vertex groups have at most one end. We generalize a result of Papasoglu--Whyte to tdlc groups and show that $G$ and $H$ are quasi-isometric if and only if they have the same number of ends and every one-ended vertex group of $(\mathcal{G}, \mathcal{A})$ is quasi-isometric to a one-ended vertex group of $(\mathcal{H}, \mathcal{B})$, and vice versa. As an application, we construct uncountably many pairwise non-quasi-isometric compactly generated non-discrete simple tdlc groups, strengthening a result by Smith.