Illinois Journal of Mathematics

Local-to-global rigidity of Bruhat–Tits buildings

Mikael De La Salle and Romain Tessera

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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

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Mathematical Reviews number (MathSciNet)

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. doi:10.1215/ijm/1506067284.

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  • 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.