Skip to main content Accessibility help
×
Home

Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

  • Thomas Delzant (a1) and Pierre Py (a2)

Abstract

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat
      Available formats
      ×

Copyright

References

Hide All
[Akh95]Akhiezer, D. N., Lie group actions in complex analysis, Aspects of Mathematics, vol. E27 (Friedr. Vieweg & Sohn, Braunschweig, 1995).
[ABCKT96]Amoros, J., Burger, M., Corlette, K., Kotschick, D. and Toledo, D., Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, vol. 44 (American Mathematical Society, Providence, RI, 1996).
[Bla09]Blanc, J., Sous-groupes algébriques du groupe de Cremona, Transform. Groups 14 (2009), 249285.
[BG80]Borchers, H. J. and Garber, W. D., Analyticity of solutions of the O(N) nonlinear σ-model, Comm. Math. Phys. 71 (1980), 299309.
[BFJ08]Boucksom, S., Favre, C. and Jonsson, M., Degree growth of meromorphic surface maps, Duke Math. J. 141 (2008), 519538.
[Bur10]Burger, M., Fundamental groups of Kähler manifolds and geometric group theory, Bourbaki Seminar No. 1022 (2010), Astérisque, to appear.
[BIM05]Burger, M., Iozzi, A. and Monod, N., Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. IMRN 2005 (2005), 13311369.
[Can11]Cantat, S., Sur les groupes de transformations birationnelles des surfaces, Ann. of Math. (2) 174 (2011), 299340.
[CF03]Cantat, S. and Favre, C., Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math. 561 (2003), 199235.
[CL10]Cantat, S. and Lamy, S., Normal subgroups in the Cremona group, Preprint (2010), arXiv:1007.0895.
[CMP03]Carlson, J., Müller-Stach, S. and Peters, C., Period mappings and period domains, Cambridge Studies in Advanced Mathematics, vol. 85 (Cambridge University Press, Cambridge, 2003).
[CT89]Carlson, J. and Toledo, D., Harmonic mappings of Kähler manifolds to locally symmetric spaces, Publ. Math. Inst. Hautes Études Sci. (1989), 173201.
[CS08]Corlette, K. and Simpson, C., On the classification of rank-two representations of quasiprojective fundamental groups, Compositio Math. 144 (2008), 12711331.
[Delz08]Delzant, T., Trees, valuations and the Green–Lazarsfeld set, Geom. Funct. Anal. 18 (2008), 12361250.
[Des06]Deserti, J., Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer, Int. Math. Res. Not. IMRN 2006 (2006).
[Des07]Deserti, J., Sur les sous-groupes nilpotents du groupe de Cremona, Bull. Braz. Math. Soc. (N.S.) 38 (2007), 377388.
[DM08]Diederich, K. and Mazzilli, E., Real and complex analytic sets. The relevance of Segre varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 447454.
[DU77]Diestel, J. and Uhl, J. J., Vector measures, Mathematical Surveys, vol. 15 (American Mathematical Society, Providence, RI, 1977).
[DF01]Diller, J. and Favre, C., Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 11351169.
[DZ01]Dolgachev, I. and Zhang, D.-Q., Coble rational surfaces, Amer. J. Math. 123 (2001), 79114.
[DK90]Donaldson, S. and Kronheimer, P. B., The geometry of four-manifolds, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1990).
[Fav10]Favre, C., Le groupe de Cremona et ses sous-groupes de type fini, in Seḿinaire Bourbaki, vol. 2008/2009, Astérisque 332 (2010), Exp. No. 998, 11–43.
[GT01]Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, Classics in Mathematics (Springer, Berlin, 2001).
[Gro93]Gromov, M., Asymptotic invariants of infinite groups, in Geometric group theory, Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 182 (Cambridge University Press, Cambridge, 1993).
[GP91]Gromov, M. and Pansu, P., Rigidity of lattices: an introduction, in Geometric topology: recent developments (Montecatini Terme, 1990), Lecture Notes in Mathematics, vol. 1504 (Springer, Berlin, 1991), 39137.
[GS92]Gromov, M. and Schoen, R., Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math. Inst. Hautes Études Sci. 76 (1992), 165246.
[JW77]Johnson, K. D. and Wallach, N. R., Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc. 229 (1977), 137173.
[JY91]Jost, J. and Yau, S. T., Harmonic maps and group representations, in Differential geometry, Pitman Monograph Surveys Pure Applied Mathematics, vol. 52 (Longman Scientific and Technical, Harlow, 1991), 241259.
[JZ00]Jost, J. and Zuo, K., Harmonic maps into Bruhat–Tits buildings and factorizations ofp-adically unbounded representations of π 1 of algebraic varieties. I, J. Algebra. Geom. 9 (2000), 142.
[Kar53]Karpelevic, F. I., Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space, Soviet. Math. Dokl. 93 (1953), 401404.
[Kli10]Klingler, B., Kaehler groups and duality, Preprint (2010), arXiv:1005.2836.
[KKM11]Klingler, B., Koziarz, V. and Maubon, J., On the second cohomology of Kähler groups, Geom. Funct. Anal. 21 (2011), 419442.
[Kna01]Knapp, A. W., Representation theory of semisimple groups. An overview based on examples, Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 2001), reprint of the 1986 original.
[KS71]Knapp, A. W. and Stein, E. M., Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489578.
[KS93]Korevaar, N. and Schoen, R., Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561659.
[KS97]Korevaar, N. and Schoen, R., Global existence theorems for harmonic maps to non-locally compact spaces, Comm. Anal. Geom. 5 (1997), 333387.
[KM58]Koszul, J.-L. and Malgrange, B., Sur certaines structures fibrées complexes, Arch. Math. (Basel) 9 (1958), 102109.
[KM08]Koziarz, V. and Maubon, J., Harmonic maps and representations of non-uniform lattices of PU(m,1), Ann. Inst. Fourier (Grenoble) 58 (2008), 507558.
[Man86]Manin, Y., Cubic forms, Algebra, Geometry, Arithmetic, Translated from the Russian by M. Hazewinkel, North-Holland Mathematical Library, vol. 4, second edition (North-Holland, Amsterdam, 1986).
[Mok88]Mok, N., Strong rigidity of irreducible quotients of polydiscs of finite volume, Math. Ann. 282 (1988), 555577.
[Mok92]Mok, N., Factorization of semisimple discrete representations of Kähler groups, Invent. Math. 110 (1992), 557614.
[Mos55]Mostow, G. D., Some new decomposition theorems for semisimple groups, Mem. Amer. Math. Soc. 1955 (1955), 3154.
[NR08]Napier, T. and Ramachandran, M., Filtered ends, proper holomorphic mappings of Kähler manifolds to Riemann surfaces and Kähler groups, Geom. Funct. Anal. 17 (2008), 16211654.
[Nis02]Nishikawa, S., Variational problems in geometry, in Iwanami series in modern mathematics, Translations of Mathematical Monographs, vol. 205 (American Mathematical Society, Providence, RI, 2002).
[Rez02]Reznikov, A., The structure of Kähler groups, I. Second cohomology, in Motives, polylogarithms and Hodge theory, part II (Irvine, CA, 1998), International Press Lecture Series, vol. 3, II (International Press, Sommerville, MA, 2002), 718730.
[Ros56]Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401443.
[Sal67]Sally, P. J., Analytic continuation of the irreducible unitary representations of the universal covering group of , Memoirs of the American Mathematical Society, vol. 69 (American Mathematical Society, Providence, RI, 1967).
[Sal70]Sally, P. J., Intertwining operators and the representations of , J. Funct. Anal. 6 (1970), 441453.
[Sam78]Sampson, J. H., Some properties and applications of harmonic mappings, Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 211228.
[Sam86]Sampson, J. H., Applications of harmonic maps to Kähler geometry, in Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984), Contemporary Mathematics, vol. 49 (American Mathematical Society, Providence, RI, 1986), 125134.
[SU87]Schoen, R. and Uhlenbeck, K., A regularity theory for harmonic maps, J. Differential Geom. 17 (1987), 307335.
[Ste70]Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30 (Princeton University Press, Princeton, NJ, 1970).
[Vek62]Vekua, I. N., Generalized analytic functions (Pergamon Press and Addison-Wesley, Reading, MA, 1962).
[Wal71]Wallach, N. R., Application of the higher osculating spaces to the spherical principal series, J. Differential Geom. 5 (1971), 405413.
[Wei55]Weil, A., On algebraic groups of transformations, Amer. J. Math. 77 (1955), 355391.
[YiH00]Yi-Hu, Y., On non-kählerianity of non-uniform lattices in SO(n,1) (n≥4), Manuscripta Math. 103 (2000), 401407.
[YiH02]Yi-Hu, Y., Non-kählerianity of non-uniform lattices in SO(3,1), Acta Math. Sin. (Engl. Ser.) 18 (2002), 801802.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed