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Article Dans Une Revue Journal of topology Année : 2019

Characterizing a vertex-transitive graph by a large ball

Résumé

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. $PGL(5,Z)$) 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.
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Dates et versions

hal-01219529 , version 1 (06-02-2024)

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Mikael de La Salle, Romain Tessera, Jean-Claude Sikorav. Characterizing a vertex-transitive graph by a large ball. Journal of topology, 2019, 12 (3), pp.705-743. ⟨10.1112/topo.12095⟩. ⟨hal-01219529⟩
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