Characterizing a vertex-transitive graph by a large ball

TL;DR

This paper investigates the local-to-global rigidity property in vertex-transitive graphs, establishing it for certain classes like Cayley graphs of specific groups, and providing counterexamples and classifications of such graphs.

Contribution

It proves local-to-global rigidity for a broad class of vertex-transitive graphs and constructs examples of graphs that do not satisfy this property, answering open questions.

Findings

Local-to-global rigidity holds for Cayley graphs of torsion-free lattices in simple Lie groups.

Counterexamples are provided among Cayley graphs of finitely presented groups like SL(4,Z).

A continuum of non-isometric large-scale simply connected vertex-transitive graphs is constructed.

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.