×

Surface group representations with maximal Toledo invariant. (English) Zbl 1035.32013

Let \(\mathbb H\) be the upper half plane and \(\Gamma_g\subset \text{SL} (2,\mathbb R)\) the fundamental group of a compact Riemann surface \(S\) of genus \(g\geq 2\) and \(X\) a Hermitian symmetric space of noncompact type, equipped with its Bergman metric and Kähler form \(\omega_X\). Let \(\rho\) be a representation of \(\Gamma_g\) into the identity component \(G(X)\) of the group of isometries of \(X\). For a smooth map \(f:\mathbb H\rightarrow X\) which is equivariant with respect to \(\rho\), the integral \(\tau_{\rho}\) of \(f^*(\omega_X)\) over \(S\) is called the Toledo invariant of \(\rho\).
It is known that \(| \tau_\rho | \leq 4\pi (g-1)r_X\), where \(r_X=\) rank \(X\). The authors are interested in the classification of representations \(\rho\) with maximal Toledo invariant \(| \tau_{\rho}| = 4\pi (g-1)r_X\). For those representations they prove: The Zariski-closure \(L\) of \(\rho (\Gamma_g)\) is a reductive subgroup of \(G(X)\), and the symmetric subspace \(Y\) of \(X\) associated to \(L\) is isometric to a tube type domain. Moreover \(\Gamma_g\) acts on \(Y\) properly discontinuously without fixed points.
An example shows that \(Y\) is not necessarily holomorphically embedded into \(X\). On the other hand, for every \(g\geq 2\) and every Hermitian symmetric space \(X\) of tube type there exists a representation \(\rho\) of some \(\Gamma_g\) into \(G(X)\) with maximal Toledo invariant and Zariski dense image.
The proof of the theorem is heavily based on former results of the authors [see M. Burger and A. Iozzi, Geom. Funct. Anal. 12, 281–292 (2002; Zbl 1006.22011), M. Burger and N. Monod, ibid., 219–280 (2002; Zbl 1006.22010) and A. Iozzi Rigidity in dynamics and geometry, Cambridge 2000, Berlin, Springer, 237–260 (2002; Zbl 1012.22023)].

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
22E40 Discrete subgroups of Lie groups
22E41 Continuous cohomology of Lie groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Borel, A.; Tits, J., Eléments unipotents et sous-groupes paraboliques de groupes réductifs, I, Invent. Math., 12, 95-104 (1971) · Zbl 0238.20055
[2] S.B. Bradlow, O. Garcia-Prada, P.B. Gothen, Surface group representations, Higgs bundles, and holomorphic triples, Preprint, 2002, http://arxiv.org/abs/math.AG/0206012; S.B. Bradlow, O. Garcia-Prada, P.B. Gothen, Surface group representations, Higgs bundles, and holomorphic triples, Preprint, 2002, http://arxiv.org/abs/math.AG/0206012 · Zbl 1070.53054
[3] Burger, M.; Iozzi, A., Boundary maps in bounded cohomology, Geom. Funct. Anal., 12, 281-292 (2002) · Zbl 1006.22011
[4] M. Burger, A. Iozzi, Bounded Kähler class rigidity of actions on Hermitian symmetric spaces, Preprint, 2002, http://www.math.ethz.ch/ iozzi/supq.ps; M. Burger, A. Iozzi, Bounded Kähler class rigidity of actions on Hermitian symmetric spaces, Preprint, 2002, http://www.math.ethz.ch/ iozzi/supq.ps
[5] Burger, M.; Monod, N., Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., 12, 219-280 (2002) · Zbl 1006.22010
[6] Clerc, J. L.; Ørsted, B., The Maslov index revisited, Transformation Groups, 6, 303-320 (2001) · Zbl 1078.53076
[7] J.L. Clerc, B. Ørsted, The Gromov norm of the Kähler class and the Maslov index, Preprint, 2002; J.L. Clerc, B. Ørsted, The Gromov norm of the Kähler class and the Maslov index, Preprint, 2002
[8] Domic, A.; Toledo, D., The Gromov norm of the Kähler class of symmetric domains, Math. Ann., 276, 425-432 (1987) · Zbl 0595.53061
[9] W.M. Goldman, Discontinuous groups and the Euler class, Thesis, University of California at Berkeley, 1980; W.M. Goldman, Discontinuous groups and the Euler class, Thesis, University of California at Berkeley, 1980
[10] Hernàndez Lamoneda, L., Maximal representations of surface groups in bounded symmetric domains, Trans. Amer. Math. Soc., 324, 405-420 (1991) · Zbl 0733.32024
[11] Ihara, S., Holomorphic imbeddings of symmetric domains, J. Math. Soc. Japan, 19, 3 (1967) · Zbl 0159.11102
[12] Iozzi, A., Bounded cohomology, boundary maps, and representations into \(Homeo_+(S^1)\) and SU \((1,n)\), (Rigidity in Dynamics and Geometry, Cambridge, UK, 2000 (2000), Springer-Verlag: Springer-Verlag Heidelberg), 237-260 · Zbl 1012.22023
[13] Monod, N., Continuous bounded cohomology of locally compact groups, (Lecture Notes in Math., 1758 (2001), Springer-Verlag: Springer-Verlag Heidelberg) · Zbl 0967.22006
[14] Satake, I., Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J. Math., 87, 425-461 (1965) · Zbl 0144.08202
[15] Toledo, D., Representations of surface groups in complex hyperbolic space, J. Differential Geom., 29, 125-133 (1989) · Zbl 0676.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.