## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 60, Number 3-4 (2016), 641-654.

Dates
Revised: 22 February 2017
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.ijm/1506067284

Digital Object Identifier
doi:10.1215/ijm/1506067284

Mathematical Reviews number (MathSciNet)
MR3707639

Zentralblatt MATH identifier
1376.20033

#### Citation

De La Salle, Mikael; Tessera, Romain. Local-to-global rigidity of Bruhat–Tits buildings. Illinois J. Math. 60 (2016), no. 3-4, 641--654. doi:10.1215/ijm/1506067284. https://projecteuclid.org/euclid.ijm/1506067284

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