Illinois Journal of Mathematics

Local-to-global rigidity of Bruhat–Tits buildings

Mikael De La Salle and Romain Tessera

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Received: 9 December 2015
Revised: 22 February 2017
First available in Project Euclid: 22 September 2017

Permanent link to this document

Zentralblatt MATH identifier

Primary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]


De La Salle, Mikael; Tessera, Romain. Local-to-global rigidity of Bruhat–Tits buildings. Illinois J. Math. 60 (2016), no. 3-4, 641--654.

Export citation


  • I. Benjamini, Coarse geometry and randomness, Lecture Notes in Mathematics, vol. 2100, Springer, Cham, 2013. Lecture notes from the 41st Probability Summer School held in Saint-Flour, 2011.
  • I. Benjamini and D. Ellis, The structure of graphs which are locally indistinguishable from a lattice. Preprint.
  • A. Borel and G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 119–171.
  • D. I. Cartwright, A. M. Mantero, T. Steger and A. Zappa, Groups acting simply transitively on the vertices of a building of type $A_{2}$ I, Geom. Dedicata 47 (1993), no. 2, 143–166.
  • D. I. Cartwright and T. Steger, A family of $\tilde{A}_{n}$-groups, Israel J. Math. 103 (1998), 125–140.
  • Y. Cornulier and P. de la Harpe, Metric geometry of locally compact groups. 228 pp. Preprint. Available at \arxivurlarXiv:1403.3796v3.
  • M. de la Salle and R. Tessera. Characterizing a vertex-transitive graph by a large ball.
  • P. Deligne, Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 119–157.
  • O. Teichmüller, Der Elementarteilersatz für nichtkommutative Ringe, S. Ber. Preuss. Akad. Wiss. (1937), 169–177.
  • A. Furman, Mostow–Margulis rigidity with locally compact targets, Geom. Funct. Anal. 11 (2001), no. 1, 30–59.
  • A. Georgakopoulos, On covers of graphs by Cayley graphs. Preprint.
  • M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory II, number 182 in LMS lecture notes (G. Niblo and M. Roller, eds.), Cambridge University Press, Cambridge, 1993.
  • H. Hasse, Theory of cyclic algebras over an algebraic number field, Trans. Amer. Math. Soc. 34 (1932), no. 1, 171–214.
  • D. Kazhdan, Representations of groups over close local fields, J. Anal. Math. 47 (1986), 175–179.
  • B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 115–197.
  • M. Krasner, Théorie non abélienne des corps de classes pour les extensions finies et séparables des corps valués complets: Approximation des corps de caractéristique $p\neq0$ par ceux de caractéristique $0$; modifications de la théorie, C. R. Acad. Sci. Paris 224 (1947), 434–436.
  • B. Krön and R. G. Möller, Quasi-isometries between graphs and trees, J. Combin. Theory Ser. B 98 (2008), 994–1013.
  • A. M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer, New York, 2000.
  • M. Ronan, Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press, Inc., Boston, MA, 1989.
  • J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, vol. 386, Springer, Berlin-New York, 1974.
  • J. Tits, A local approach to buildings, The Geometric Vein (Coxeter Festschrift), Springer, New York, 1981, pp. 317–322.
  • J. Tits, Immeubles de type affine, Buildings and the geometry of diagrams (Como, 1984), Lecture Notes in Math., vol. 1181, Springer, Berlin, 1986, pp. 159–190.
  • V. I. Trofimov, Graphs with polynomial growth, Math. USSR, Sb. 51 (1985), no. 2, 405–417.
  • A. Weil, Basic number theory, 3rd ed., Die Grundlehren der Mathematischen Wissenschaften, vol. 144, Springer, New York-Berlin, 1974.