Skip to main content
SpringerLink
Log in

On the first cohomology of cocompact arithmetic groups

  • Published:
Proceedings of the Indian Academy of Sciences - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Results of Matsushima and Raghunathan imply that the first cohomology of a cocompact irreducible lattice in a semisimple Lie groupG, with coefficients in an irreducible finite dimensional representation ofG, vanishes unless the Lie group isSO(n, 1) orSU(n, 1) and the highest weight of the representation is an integral multiple of that of the standard representation.

We show here that every cocompact arithmetic lattice inSO(n, 1) contains a subgroup of finite index whose first cohomology is non-zero when the representation is one of the exceptional types mentioned above.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [B-W] Borel A, Wallach N, Continuous cohomology, discrete subgroups, and representations of reductive groups,Ann. Math. Stud. (1980) (Princeton University Press)

  • [K1] Kazhdan D A, Connection between the dual space of a group and the structure of its closed subgroups,Funct. Anal. Appl. 1 (1977) 63–65

    Article  Google Scholar 

  • [K2] Kazhdan D A, Some applications of the Weil representation,J. Anal. 32 (1977) 235–248

    MATH  MathSciNet  Google Scholar 

  • [Kos] Kostant B, On the existence and irreducibility of certain representations, Lie groups and representations (1971) (Budapest: Summer School of the Bolyai Janos Math. Society)

    Google Scholar 

  • [L] Li J-S, Non-vanishing theorems for the cohomology of certain arithmetic quotients,J. Reine Angew. Math. 428 (1992) 177–217

    MATH  MathSciNet  Google Scholar 

  • [L-M] Li J-S, Millson J, On the first Betti-number of a hyperbolic manifold with arithmetic fundamental group, Preprint

  • [Ma] Matsushima Y, On the first Betti number of compact quotients of higher dimensional symmetric spaces,Ann. Math. 75 (1962) 312–330

    Article  MathSciNet  Google Scholar 

  • [M1] Millson J, On the first Betti-number of a constant negatively curved manifold,Ann. Math. 104 (1976) 235–247

    Article  MathSciNet  Google Scholar 

  • [M2] Millson J, On the first cohomology of arithmetic groups,Topology 24 (1985) 495–498

    Article  MATH  MathSciNet  Google Scholar 

  • [Pr-R] Prasad G, Raghunathan M S, On the congruence subgroup problem: determination of the metaplectic kernel,Invent. Math. 71 (1983) 21–42

    Article  MATH  MathSciNet  Google Scholar 

  • [R1] Raghunathan M S, On the first cohomology of discrete subgroups,Am. J. Math. 87 (1965) 102–139

    Article  MathSciNet  Google Scholar 

  • [R-V] Raghunathan M S, Venkataramana T N, The first Betti-number of arithmetic groups and the congruence subgroup problem, Proceedings of a Conference held in honour of R Steinberg, UCLA, 1992

  • [T] Tits J, Classification of algebraic semi-simple groups,Proc. Symp. Pure Math. (1966) (Boulder: Colorado, 1965) p. 33–62

  • [To] Tomanov G, Projective simplicity of groups of rational points, Banach Center Publ. (Topics in Algebra) Part II26 (1985) 455–466

    MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. T N Venkataramana

    Present address: International Centre for Theoretical Physics, P.O. Box 586, 43100, Trieste, Italy

Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, 400 005, Mumbai, India

    T N Venkataramana

Authors
  1. T N Venkataramana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T N Venkataramana.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Venkataramana, T.N. On the first cohomology of cocompact arithmetic groups. Proc. Indian Acad. Sci. (Math. Sci.) 106, 245–259 (1996). https://doi.org/10.1007/BF02867433

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02867433

Keywords

Search

Navigation