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Geometric and Functional Analysis

, Volume 19, Issue 3, pp 678–721 | Cite as

Tight Homomorphisms and Hermitian Symmetric Spaces

  • Marc Burger
  • Alessandra IozziEmail author
  • Anna Wienhard
Article

Abstract

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore, we classify all tight embeddings of the Poincaré disk.

Keywords and phrases

Hermitian symmetric spaces bounded cohomology Shilov boundary totally geodesic embedding tight homomorphism 

2000 Mathematics Subject Classification

53C24 53C35 32M15 

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References

  1. BGSS.
    Breuillard E., Gelander T., Souto J., Storm P. (2006) Dense embeddings of surface groups. Geom. Topol. 10: 1373–1389 (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  2. BuIo1.
    Burger M., Iozzi A. (2004) Bounded Kähler class rigidity of actions on Hermitian symmetric spaces. Ann. Sci. École Norm. Sup. (4) 37(1): 77–103zbMATHMathSciNetGoogle Scholar
  3. BuIo2.
    Burger M., Iozzi A. (2007) Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity. Pure Appl. Math. Q. 125(1): 1–23zbMATHMathSciNetGoogle Scholar
  4. BuIoLW.
    Burger M., Iozzi A., Labourie F., Wienhard A. (2005) Maximal representations of surface groups: Symplectic Anosov structures, Special Issue: In Memory of Armand Borel, Part 2 of 3. Quarterly Journal of Pure and Applied Mathematics 1(3): 555–601MathSciNetGoogle Scholar
  5. BuIoW1.
    M. Burger, A. Iozzi, A. Wienhard, Surface group representations with maximal Toledo invariant, Annals of Math., to appear; http://www.arXiv.org/math.DG/0605656.
  6. BuIoW2.
    Burger M., Iozzi A., Wienhard A. (2003) Surface group representations with maximal Toledo invariant. C. R. Acad. Sci. Paris, Sér. I 336: 387–390zbMATHMathSciNetGoogle Scholar
  7. BuIoW3.
    Burger M., Iozzi A., Wienhard A. (2007) Hermitian symmetric spaces and Kähler rigidity. Transform. Groups 12(1): 5–32zbMATHCrossRefMathSciNetGoogle Scholar
  8. BuM1.
    Burger M., Monod N. (1999) Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1(2): 199–235zbMATHCrossRefMathSciNetGoogle Scholar
  9. BuM2.
    Burger M., Monod N. (2002) Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12: 219–280zbMATHCrossRefMathSciNetGoogle Scholar
  10. CØ.
    Clerc J.L., Ørsted B. (2003) The Gromov norm of the Kaehler class and the Maslov index. Asian J. Math. 7(2): 269–295zbMATHMathSciNetGoogle Scholar
  11. DT.
    Domic A., Toledo D. (1987) The Gromov norm of the Kaehler class of symmetric domains. Math. Ann. 276(3): 425–432zbMATHCrossRefMathSciNetGoogle Scholar
  12. Dy.
    Dynkin E.B. (1957) Semisimple subalgebras of semisimple Lie algebras. Am. Math. Soc., Transl. Ser. II, AMS 6: 111–243zbMATHGoogle Scholar
  13. E.
    van Est W.T. (1953) Group cohomology and Lie algebra cohomology in Lie groups, I, II, Nederl. Akad. Wetensch. Proc. Series A 56=Indag. Math. 15: 484–504Google Scholar
  14. I1.
    Ihara S.-I. (1966) Holomorphic imbeddings of symmetric domains into a symmetric domain. Proc. Japan Acad. 42: 193–197zbMATHCrossRefMathSciNetGoogle Scholar
  15. I2.
    Ihara S.-I. (1967) Holomorphic imbeddings of symmetric domains. J. Math. Soc. Japan 19: 261–302zbMATHMathSciNetCrossRefGoogle Scholar
  16. Io.
    A. Iozzi, Bounded cohomology, boundary maps, and representations into Homeo+(S 1) and SU(1, n), in “Rigidity in Dynamics and Geometry, Cambridge, UK, 2000”, Springer Verlag (2002), 237–260.Google Scholar
  17. KW.
    Korányi A., Wolf J.A. (1965) Realization of Hermitian symmetric spaces as generalized half-planes. Ann. of Math. (2) 81: 265–288CrossRefMathSciNetGoogle Scholar
  18. KoM.
    V. Koziarz, J. Maubon, Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type, preprint; http://www.arXiv.org/math/0703174.
  19. L.
    O. Loos, Bounded symmetric domains and Jordan pairs, Department of Mathematics, University of California at Irvine, Mathematical Lectures, 1977.Google Scholar
  20. M.
    N. Monod, Continuous Bounded Cohomology of Locally Compact Groups, Springer Lecture Notes in Math. 1758 (2001).Google Scholar
  21. S1.
    Satake I. (1960) On representations and compactifications of symmetric Riemannian spaces. Ann. of Math. (2) 71: 77–110CrossRefMathSciNetGoogle Scholar
  22. S2.
    I. Satake, Algebraic Structures of Symmetric Domains, Kanô Memorial Lectures 4, Iwanami Shoten, Tokyo, 1980.Google Scholar
  23. V.
    È.B. Vinberg (ed.), Lie groups and Lie algebras, III, Structure of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences 41, Springer-Verlag, Berlin (1994); Current Problems in Mathematics. Fundamental Directions 41 (in Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1990).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.FIM, ETH ZentrumZürichSwitzerland
  2. 2.Department MathematikETH ZentrumZürichSwitzerland
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

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