Characterizing a vertex-transitive graph by a large ball

Abstract

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. Answering a question of Cornulier, we also construct a continuum of non pairwise isometric large-scale simply connected locally finite vertex-transitive graphs. This question was motivated by the fact that large-scale simply connected Cayley graphs are precisely Cayley graphs of finitely presented groups and therefore have countably many isometric classes.

v1: 38 pages. With an Appendix by Jean-Claude Sikorav v2: 48 pages. Several improvements in the presentation. To appear in Journal of Topology

References in corpus (2)

• Metric geometry of locally compact groups
• Transitive graphs uniquely determined by their local structure

Cited by in corpus (4)

• Cayley graphs with few automorphisms: the case of infinite groups
• Cayley graphs with few automorphisms
• Local-to-Global-rigidity of lattices in $S{L}_n(\mathbb{K})$
• Fundamental groups and group presentations with bounded relator lengths