  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 non-geometric
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
non-elementary 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.
