Cayley–Abels graph

Summary

Let $G$ be a compactly generated totally disconnected locally compact group. A Cayley–Abels graph of $G$ is a connected locally finite graph on which $G$ acts vertex-transitively with compact open vertex stabilisers. It generalises the Cayley graph of a finitely generated group, which is the case of a discrete group. Every such $G$ has a Cayley–Abels graph, and any two of them are quasi-isometric. So $G$ has a well-defined large-scale geometry, as in geometric group theory.