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 $d\geq4$. 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 $\mathrm{SL}(d,\mathbf{F}_{p}(\!(t)\!))$ is not local-to-global rigid, while the Bruhat–Tits building of $\mathrm{SL}(d,\mathbf{Q}_{p})$ is local-to-global rigid.