Skip to main content
Log in

A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We give a characterisations for an abstract group to be an S-arithmetic group of a higher rank semi-simple group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • [B-M-S] Bass, H., Milnor, J. and Serre, J.-P.: Solution of the congruence subgroup problem for SLn(n ⩳ 3) and SP2n(n ⩳ 2), Publ. Math. IHES 33 (1967), 59–137.

    Google Scholar 

  • [B-H] Borel, A. and Harish-Chandra, S.: Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485–535.

    Google Scholar 

  • [B-S] Borel, A. and Serre, J.-P.: Cohomologie d'immeubles et de Groupes S-arithmetiques, Topology 15 (1976), 211–232.

    Google Scholar 

  • [B-T] Borel, A. and Tits, J.: Groupes reductifs, Publ Math. I.H.E.S. 27 (1965), 55–150; Compliments a' l'article groupes reductifs, Publ. Math. I.H.E.S. 41 (1972), 253–276.

    Google Scholar 

  • [C-F] Cassels, S. W. and Frolich, A.: Algebraic Number Theory, Academic Press, London, 1967.

    Google Scholar 

  • [La] Lazard: Groupes analitiques p-adiques, Publ. Math. I.H.E.S. (26) (1965), 5–219.

    Google Scholar 

  • [Lu] Lubotzky, A.: Dimension Functions for Discrete Groups, Proc. of Groups, St. Andrews (1985), London Math. Soc., Lecture Notes Ser. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 254–262.

    Google Scholar 

  • [L-M-R1] Lubotzky, A., Mozes, S. and Raghunathan, M. S.: Cyclic subgroups of exponential growth and metrics on discrete groups, C.R. Acad. Sci. Paris Ser I, Math. 317(8) (1993), 735–740.

    Google Scholar 

  • [L-M-R2] Lubotzky, A., Mozes, A. and Raghunathan, M. S.: The word and Riemannian metrics on lattices in semi-simple groups, Publ. Math. I.H.E.S., to appear.

  • [M] Margulis, G. A.: Discrete Subgroups of Semi-Simple Lie Groups, Ergebnisse der Math. (3) 17, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  • [M2] Margulis, G. A.: Finiteness of quotients of discrete subgroups, Funktsional Anal. i Priolozhen. 13(3) (1979), 28–39.

    Google Scholar 

  • [P] Platonov, V. P.: The strong approximation problem and the Kneser — Tits conjecture for algebraic groups, Izv. Akad.Math SSSR, Serié 33(6) (1969), 1211–1219.

    Google Scholar 

  • [P-R] Platonov, V. P. and Rapinchuk, A. S.: Algebraic Groups and N umber Theory, Academic Press, Boston, 1994.

    Google Scholar 

  • [R] Raghunathan, M. S.: Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. Springer, Berlin, 1972.

    Google Scholar 

  • [R1] Raghunathan, M. S.: On the congruence subgroup problem, Publ.Math. I.H.E.S. 46 (1176), 107–161.

    Google Scholar 

  • [R2] Raghunathan, M. S.: On the congruence subgroup problem II, Invent.Math. 85(1) (1986), 73–117.

    Google Scholar 

  • [R3] Raghunathan, M. S.: Generators for arithmetic groups, Pacific J. Math. 152(2) (1992), 365–373.

    Google Scholar 

  • [R4] Raghunathan, M. S.: The congruence subgroup problem, In: Proc. Hyderaband Conf. on Algebraic Groups, Hyderaband, 1989, pp. 465–494.

  • [S1] Serre, J.-P.: Lie Groups and Lie Algebras, Benjamin, New York, 1965.

    Google Scholar 

  • [S2] Serre, J.-P.: Cohomologie des groupes discrete, In: Prospects in Math., Ann. of Math. Stud. 70, Princeton Univ. Press, 1971, pp. 77–169.

    Google Scholar 

  • [T] Tits, J.: Classification of Algebraic Semi-simple Groups, in Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math. 9 Amer. Math. Soc., Providence, 1966.

    Google Scholar 

  • [Ve 1] Venkataramana, T. N.: Systems of generators for arithmetic groups, Pac. J. Math 166(1) (1994), 193–212.

    Google Scholar 

  • [V] Vinberg, E. B.: Rings of definition of dense subgroups of semisimple Lie groups, Izv. AkadNauk SSSR Ser. Math 35 (1971), 45–55.

    Google Scholar 

  • [W] Weil, A.: Adeles and AlgebraicGroups, Progr. inMath. 23, Birkhauser, Basel, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lubotzky, A., Venkataramana, T.N. A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups. Geometriae Dedicata 90, 1–28 (2002). https://doi.org/10.1023/A:1014985620796

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014985620796

Navigation