Local-to-global rigidity of Bruhat-Tits buildings

TL;DR

This paper investigates the local-to-global rigidity property of the 1-skeletons of affine Bruhat-Tits buildings, showing it depends on the characteristic of the underlying field, with implications for certain algebraic groups.

Contribution

It establishes a precise criterion for local-to-global rigidity of Bruhat-Tits buildings based on the characteristic of the underlying field.

Findings

Buildings over characteristic zero fields are local-to-global rigid.

Buildings over fields with positive characteristic are not local-to-global rigid.

The rigidity property varies significantly between different algebraic groups.

Abstract

A vertex-transitive graph X is called local-to-global rigid if there exists R such that every other graph whose balls of radius R are isometric to the balls of radius R in X is covered by X. Let $d\geq 4$. We show that the 1-skeleton of an affine Bruhat-Tits building of type $\widetilde A_{d-1}$ is local-to-global rigid if and only if the underlying field has characteristic 0. For example the Bruhat-Tits building of $SL(d,F_p((t)))$ is not local-to-global rigid, while the Bruhat-Tits building of $SL(d,Q_p)$ is local-to-global rigid.