Graphically discrete groups and rigidity

TL;DR

The paper introduces graphical discreteness as a robust rigidity-enforcing property for finitely generated groups, showing that many classic groups (nilpotent, lattices in semisimple Lie groups, and closed 3-manifold groups) are graphically discrete and thus action rigid on common spaces. It develops a comprehensive framework linking graph-of-spaces decompositions, common-cover theorems, and lattice-embedding analysis to deduce both action rigidity and, in many hyperbolic settings, quasi-isometric rigidity. Key results include: (i) free products of one-ended graphically discrete groups are action rigid within virtually torsion-free groups; (ii) hyperbolic graphs of manifold groups with almost malnormal quasi-convex edge groups satisfy quasi-isometric rigidity via common-model geometries; and (iii) graphical discreteness is not a commensurability invariant, with explicit counterexamples. The work also provides a versatile toolkit—blowups, common covers, and tree-of-spaces constructions—that unifies approaches from lattice envelopes, pattern rigidity, and cubulation to obtain strong rigidity statements. Overall, the paper advances understanding of when geometric actions force algebraic rigidity, and clarifies the roles of vertex-group discreteness, edge-group malnormality, and boundary dynamics in governing quasi-isometric equivalence classes of groups.

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 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.

Figures (2)

• Figure 1: The automorphism group of the graph $X$ is not discrete. The vertices are grouped in blue $\mathop{\mathrm{Aut}}\nolimits(X)$-invariant finite equivalence classes so that the action of $\mathop{\mathrm{Aut}}\nolimits(X)$ on the quotient graph $X/\sim$ is discrete.
• Figure 2: An example where $\mathop{\mathrm{Aut}}\nolimits(X)$ acts discretely on $X/\sim$, but the group $\mathop{\mathrm{Aut}}\nolimits(X/\sim)$ is not discrete. On the left, each set of colored vertices indicates one equivalence class, and each black vertex is in its own equivalence class.

Theorems & Definitions (241)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4