Graphically discrete groups and rigidity

Abstract We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact‐by‐discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3‐manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one‐ended virtually torsion‐free graphically discrete groups are action rigid within the class of virtually torsion‐free groups. We also prove quasi‐isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi‐convex subgroups. This includes the case of two hyperbolic 3‐manifold groups amalgamated along a quasi‐convex malnormal non‐abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.