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

TL;DR

This paper investigates the local-to-global rigidity of certain graphs related to lattices in special linear groups over local fields, extending known rigidity properties to broader classes of graphs and establishing new local structural results.

Contribution

It extends local-to-global rigidity from Bruhat-Tits buildings to graphs quasi-isometric to these buildings, including torsion-free lattices of SL_n(K).

Findings

Vertices are uniquely determined by neighboring vertices in a fixed orbit.

The paper proves a local structure property of the building.

Extension of rigidity to a broader class of graphs.

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 $PSL_n(\mathbb{K})$ with $n\geq 4$ 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 $SL_n(\mathbb{K})$. The demonstration is the occasion to prove a result on the local structure of the building. We show that if we fix a $PSL_n(\mathbb{K})$-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.