Harmonic maps and representations of non-uniform lattices of PU (m,1)
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 507-558.

We study representations of lattices of PU (m,1) into 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 PU (n,1) of non-uniform lattices in 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.

Nous étudions les représentations des réseaux de PU (m,1) dans 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 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.

DOI: 10.5802/aif.2359
Classification: 22E40, 32Q05, 32Q20, 53C24, 53C35, 53C43
Keywords: Representations, non-uniform lattices, complex hyperbolic space, Toledo invariant, harmonic maps, surfaces of finite topological type, rigidity
Mot clés : représentations, réseaux non uniformes, espace hyperbolique complexe, invariant de Toledo, applications harmoniques, surfaces de type topologique fini, rigidité

Koziarz, Vincent 1; Maubon, Julien 1

1 Université Henri Poincaré Institut Elie Cartan BP 239 54506 Vandœuvre-lès-Nancy Cedex (France)
@article{AIF_2008__58_2_507_0,
     author = {Koziarz, Vincent and Maubon, Julien},
     title = {Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$},
     journal = {Annales de l'Institut Fourier},
     pages = {507--558},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     doi = {10.5802/aif.2359},
     mrnumber = {2410381},
     zbl = {1147.22009},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2359/}
}
TY  - JOUR
AU  - Koziarz, Vincent
AU  - Maubon, Julien
TI  - Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 507
EP  - 558
VL  - 58
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2359/
DO  - 10.5802/aif.2359
LA  - en
ID  - AIF_2008__58_2_507_0
ER  - 
%0 Journal Article
%A Koziarz, Vincent
%A Maubon, Julien
%T Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$
%J Annales de l'Institut Fourier
%D 2008
%P 507-558
%V 58
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2359/
%R 10.5802/aif.2359
%G en
%F AIF_2008__58_2_507_0
Koziarz, Vincent; Maubon, Julien. Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 507-558. doi : 10.5802/aif.2359. https://aif.centre-mersenne.org/articles/10.5802/aif.2359/

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

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

[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., Volume 12 (2002) no. 2, pp. 219-280 | DOI | MR | Zbl

[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 | DOI | EuDML | Numdam | MR | Zbl

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

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

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

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

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

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

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

[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 | DOI | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

[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, Pure and Applied Mathematics, 2, Marcel Dekker, New York, 1970 | MR | Zbl

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

[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 | Zbl

[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, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991 | MR | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

[40] Zucker, Steven L 2 cohomology of warped products and arithmetic groups, Invent. Math., Volume 70 (1982/83) no. 2, pp. 169-218 | DOI | MR | Zbl

Cited by Sources: