Local-to-global rigidity of Bruhat-Tits buildings

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.