Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$

[Applications harmoniques et représentations des réseaux non uniformes de ${\rm PU}(m,1)$]

Koziarz, Vincent ; Maubon, Julien

Annales de l'Institut Fourier, Tome 58 (2008), p. 507-558 / Harvested from Numdam

Résumé

Nous étudions les représentations des réseaux de ${\rm PU}(m,1)$ dans ${\rm PU}(n,1)$. Nous montrons que si la représentation est réductive et si $m$ est supérieur ou égal à 2, il existe une application équivariante harmonique d’énergie finie de l’espace hyperbolique complexe de dimension $m$ dans l’espace hyperbolique complexe de dimension $n$. Ceci nous permet de donner une preuve géométrique de résultats de rigidité obtenus par M. Burger et A. Iozzi. Nous définissons aussi un nouvel invariant associé aux représentations dans ${\rm PU}(n,1)$ des groupes fondamentaux des surfaces orientables de type topologique fini et de caractéristique d’Euler négative. Nous montrons que cet invariant est borné par une constante dépendant uniquement de la caractéristique d’Euler de la surface et nous donnons une caractérisation complète des représentations d’invariant maximal, généralisant ainsi les résultats de D. Toledo sur les surfaces compactes.

Abstract

We study representations of lattices of ${\rm PU}(m,1)$ into ${\rm PU}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into ${\rm PU}(n,1)$ of non-uniform lattices in ${\rm PU}(1,1)$, and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.

Détail

Publié le : 2008-01-01

MR 2410381 | Zbl 1147.22009

DOI : https://doi.org/10.5802/aif.2359
Classification:  22E40,  32Q05,  32Q20,  53C24,  53C35,  53C43
Mots clés: représentations, réseaux non uniformes, espace hyperbolique complexe, invariant de Toledo, applications harmoniques, surfaces de type topologique fini, rigidité

Bibliographie

[1] Auslander, Louis Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. (2), Tome 71 (1960), pp. 579-590 | Article | MR 121423 | Zbl 0099.25602

[2] Biquard, Olivier Métriques d’Einstein à cusps et équations de Seiberg-Witten, J. Reine Angew. Math., Tome 490 (1997), pp. 129-154 | Article | MR 1468928 | Zbl 0891.53029

[3] Burger, M.; Iozzi, A. Bounded cohomology and representation varieties of lattices in PSU$(1,n)$ (2001) (Preprint)

[4] Burger, M.; Iozzi, A. Letter (2003)

[5] Burger, M.; Monod, N. Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., Tome 12 (2002) no. 2, pp. 219-280 | Article | MR 1911660 | Zbl 1006.22010

[6] Carlson, James A.; Toledo, Domingo Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. (1989) no. 69, pp. 173-201 | Article | Numdam | MR 1019964 | Zbl 0695.58010

[7] Corlette, Kevin Flat $G$-bundles with canonical metrics, J. Differential Geom., Tome 28 (1988) no. 3, pp. 361-382 | MR 965220 | Zbl 0676.58007

[8] Corlette, Kevin Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2), Tome 135 (1992) no. 1, pp. 165-182 | Article | MR 1147961 | Zbl 0768.53025

[9] Eells, James Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Tome 86 (1964), pp. 109-160 | Article | MR 164306 | Zbl 0122.40102

[10] Gaffney, Matthew P. A special Stokes’s theorem for complete Riemannian manifolds, Ann. of Math. (2), Tome 60 (1954), pp. 140-145 | Article | Zbl 0055.40301

[11] Goldman, W. M.; Millson, J. J. Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math., Tome 88 (1987) no. 3, pp. 495-520 | Article | MR 884798 | Zbl 0627.22012

[12] Goldman, William M. Representations of fundamental groups of surfaces, Geometry and topology (College Park, Md., 1983/84), Springer, Berlin (Lecture Notes in Math.) Tome 1167 (1985), pp. 95-117 | MR 827264 | Zbl 0575.57027

[13] Goldman, William M. Complex hyperbolic geometry, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1999) (Oxford Science Publications) | MR 1695450 | Zbl 0939.32024

[14] Gromov, Mikhail; Schoen, Richard Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, pp. 165-246 | Article | Numdam | MR 1215595 | Zbl 0896.58024

[15] Gusevskii, Nikolay; Parker, John R. Representations of free Fuchsian groups in complex hyperbolic space, Topology, Tome 39 (2000) no. 1, pp. 33-60 | Article | MR 1710991 | Zbl 0977.32017

[16] Gusevskii, Nikolay; Parker, John R. Complex hyperbolic quasi-Fuchsian groups and Toledo’s invariant, Geom. Dedicata, Tome 97 (2003), pp. 151-185 (Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999)) | Article | Zbl 1042.57023

[17] Helgason, Sigurdur Groups and geometric analysis, Academic Press Inc., Orlando, FL, Pure and Applied Mathematics, Tome 113 (1984) (Integral geometry, invariant differential operators, and spherical functions) | MR 754767 | Zbl 0543.58001

[18] Hernández, Luis Kähler manifolds and $1/4$-pinching, Duke Math. J., Tome 62 (1991) no. 3, pp. 601-611 | Article | MR 1104810 | Zbl 0725.53068

[19] Hummel, Christoph; Schroeder, Viktor Cusp closing in rank one symmetric spaces, Invent. Math., Tome 123 (1996) no. 2, pp. 283-307 | Article | MR 1374201 | Zbl 0860.53025

[20] Iozzi, Alessandra Bounded cohomology, boundary maps, and rigidity of representations into ${\rm Homeo}_+(\bf S^1)$ and ${\rm SU}(1,n)$, Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin (2002), pp. 237-260 | MR 1919404 | Zbl 1012.22023

[21] Johnson, Dennis; Millson, John J. Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 67 (1987), pp. 48-106 | MR 900823 | Zbl 0664.53023

[22] Jost, Jürgen; Zuo, Kang Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties, J. Differential Geom., Tome 47 (1997) no. 3, pp. 469-503 | MR 1617644 | Zbl 0911.58012

[23] Kapovich, M. On normal subgroups in the fundamental groups of complex surfaces, arXiv:math.GT/9808085

[24] Kobayashi, Shoshichi Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, Pure and Applied Mathematics, Tome 2 (1970) | MR 277770 | Zbl 0207.37902

[25] Li, Peter Complete surfaces of at most quadratic area growth, Comment. Math. Helv., Tome 72 (1997) no. 1, pp. 67-71 | Article | MR 1456316 | Zbl 1008.53034

[26] Lichnerowicz, André Applications harmoniques et variétés kähleriennes, Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69), Academic Press, London (1968/1969), pp. 341-402 | MR 262993 | Zbl 0193.50101

[27] Livné, R. On certain covers of the universal elliptic curve, Harvard University (1981) (Ph. D. Thesis)

[28] Margulis, G. A. Discrete subgroups of semisimple Lie groups, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 17 (1991) | MR 1090825 | Zbl 0732.22008

[29] Mok, Ngaiming; Siu, Yum Tong; Yeung, Sai-Kee Geometric superrigidity, Invent. Math., Tome 113 (1993) no. 1, pp. 57-83 | Article | MR 1223224 | Zbl 0808.53043

[30] Mostow, G. D. Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J. (1973) (Annals of Mathematics Studies, No. 78) | MR 385004 | Zbl 0265.53039

[31] Pansu, Pierre Sous-groupes discrets des groupes de Lie: rigidité, arithméticité, Astérisque (1995) no. 227, Exp. No. 778, 3, pp. 69-105 (Séminaire Bourbaki, Vol. 1993/94) | Numdam | MR 1321644 | Zbl 0835.22011

[32] Reznikov, Alexander G. Harmonic maps, hyperbolic cohomology and higher Milnor inequalities, Topology, Tome 32 (1993) no. 4, pp. 899-907 | Article | MR 1241878 | Zbl 0804.57013

[33] Sampson, J. H. Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4), Tome 11 (1978) no. 2, pp. 211-228 | Numdam | MR 510549 | Zbl 0392.31009

[34] Schoen, Richard; Yau, Shing Tung On univalent harmonic maps between surfaces, Invent. Math., Tome 44 (1978) no. 3, pp. 265-278 | Article | MR 478219 | Zbl 0388.58005

[35] Serre, Jean-Pierre Arbres, amalgames, ${\rm SL}_{2}$, Société Mathématique de France, Paris (1977) (Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46) | MR 476875

[36] Siu, Yum Tong The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2), Tome 112 (1980) no. 1, pp. 73-111 | Article | MR 584075 | Zbl 0517.53058

[37] Toledo, Domingo Harmonic maps from surfaces to certain Kaehler manifolds, Math. Scand., Tome 45 (1979) no. 1, pp. 13-26 | MR 567429 | Zbl 0435.58008

[38] Toledo, Domingo Representations of surface groups in complex hyperbolic space, J. Diff. Geom., Tome 29 (1989), pp. 125-133 | MR 978081 | Zbl 0676.57012

[39] Wood, John C. Holomorphicity of certain harmonic maps from a surface to complex projective $n$-space, J. London Math. Soc. (2), Tome 20 (1979) no. 1, pp. 137-142 | Article | MR 545210 | Zbl 0407.58026

[40] Zucker, Steven $L_{2}$ cohomology of warped products and arithmetic groups, Invent. Math., Tome 70 (1982/83) no. 2, pp. 169-218 | Article | MR 684171 | Zbl 0508.20020