Local-to-Global-rigidity of lattices in $S{L}_n(\mathbb{K})$

Abstract

A vertex-transitive graph $\mathcal{G}$ is called Local-to-Global rigid if there exists $R>0$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $\mathcal{G}$ is covered by $\mathcal{G}$. An example of such a graph is given by the Bruhat-Tits building of $PS{L}_n(\mathbb{K})$ with $n\ge 4$ and $\mathbb{K}$ a non-Archimedean local field of characteristic zero.. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of $S{L}_n(\mathbb{K})$. The demonstration is the occasion to prove a result on the local structure of the building. We show that if we fix a $PS{L}_n(\mathbb{K})$-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.

35 pages, 12 figures. (Changes in v3: typos corrected. .tex files now available. Published in Annales de l'institut Fourier.)

References in corpus (3)

• Characterizing a vertex-transitive graph by a large ball
• On the structure and arithmeticity of lattice envelopes
• On the structure of graphs which are locally indistinguishable from a lattice