Geometry & Topology

Quasi-isometric rigidity of higher rank $S$–arithmetic lattices

Kevin Wortman

Full-text: Open access


We show that S–arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.

Article information

Geom. Topol., Volume 11, Number 2 (2007), 995-1048.

Received: 19 November 2004
Accepted: 21 September 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20G30: Linear algebraic groups over global fields and their integers 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

quasi-isometry arithmetic quasi-isometry arithmetic


Wortman, Kevin. Quasi-isometric rigidity of higher rank $S$–arithmetic lattices. Geom. Topol. 11 (2007), no. 2, 995--1048. doi:10.2140/gt.2007.11.995.

Export citation


  • H Abels, Finiteness properties of certain arithmetic groups in the function field case, Israel J. Math. 76 (1991) 113–128
  • P Abramenko, Finiteness properties of Chevalley groups over $\mathbb{F}_q[t]$, Israel J. Math. 87 (1994) 203–223
  • H Behr, $\mathrm{SL}_3(\mathbb{F}_q[t])$ is not finitely presentable, from: “Homological group theory (Proc. Sympos., Durham, 1977)”, London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press, Cambridge (1979) 213–224
  • F Blume, Ergodic theory, from: “Handbook of measure theory, Vol. II”, North–Holland, Amsterdam (2002) 1185–1235
  • A Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966) 78–89
  • A Borel, Linear algebraic groups, second edition, Graduate Texts in Mathematics 126, Springer, New York (1991)
  • A Borel, T A Springer, Rationality properties of linear algebraic groups. II, Tôhoku Math. J. $(2)$ 20 (1968) 443–497
  • A Borel, J Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. $(2)$ 97 (1973) 499–571
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin (1999)
  • K S Brown, Buildings, Springer, New York (1989)
  • C Druţu, Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank, Geom. Funct. Anal. 10 (2000) 327–388
  • A Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, J. Amer. Math. Soc. 11 (1998) 321–361
  • A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653–692
  • B Farb, The quasi-isometry classification of lattices in semisimple Lie groups, Math. Res. Lett. 4 (1997) 705–717
  • B Farb, R Schwartz, The large-scale geometry of Hilbert modular groups, J. Differential Geom. 44 (1996) 435–478
  • G Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen III, J. Reine Angew. Math. 274/275 (1975) 125–138
  • B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997) 115–197
  • A Lubotzky, S Mozes, M S Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. 91 (2000) 5–53
  • G A Margulis, Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer, Berlin (1991)
  • G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press, Princeton, N.J. (1973)
  • V Platonov, A Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Boston (1994)
  • G Prasad, Strong rigidity of $\mathbb{Q}$–rank $1$ lattices, Invent. Math. 21 (1973) 255–286
  • G Prasad, Strong approximation for semi-simple groups over function fields, Ann. of Math. $(2)$ 105 (1977) 553–572
  • G Prasad, Lattices in semisimple groups over local fields, from: “Studies in algebra and number theory”, Adv. in Math. Suppl. Stud. 6, Academic Press, New York (1979) 285–356
  • M Ratner, On the $p$–adic and $S$–arithmetic generalizations of Raghunathan's conjectures, from: “Lie groups and ergodic theory (Mumbai, 1996)”, Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res., Bombay (1998) 167–202
  • R E Schwartz, The quasi-isometry classification of rank one lattices, Inst. Hautes Études Sci. Publ. Math. 82 (1995) 133–168 (1996)
  • R E Schwartz, Quasi-isometric rigidity and Diophantine approximation, Acta Math. 177 (1996) 75–112
  • J Taback, Quasi-isometric rigidity for $\mathrm{PSL}_2(\mathbb{Z}[1/p])$, Duke Math. J. 101 (2000) 335–357
  • J Tits, Algebraic and abstract simple groups, Ann. of Math. $(2)$ 80 (1964) 313–329
  • J Tits, Buildings of spherical type and finite BN–pairs, Lecture Notes in Mathematics 386, Springer, Berlin (1974)
  • T N Venkataramana, On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988) 255–306
  • K Wortman, Quasiflats with holes in reductive groups, Algebr. Geom. Topol. 6 (2006) 91–117
  • K Wortman, Quasi-isometries of $\mathbf{SL}_n(\mathbb{F}_q[t])$ (in preparation)