Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Maximal representations of surface groups in bounded symmetric domains


Author: Luis Hernández
Journal: Trans. Amer. Math. Soc. 324 (1991), 405-420
MSC: Primary 32M15; Secondary 22E40
MathSciNet review: 1033234
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Abstract: Let $ \Gamma$ be the fundamental group of a hyperbolic surface of genus $ g$; for $ 1 \le p \le q,PSU(p,q)$ is the group of isometries of a certain Hermitian symmetric space $ {D_{p,q}}$ of rank $ p$. There exists a characteristic number $ c:\operatorname{Hom} (\Gamma ,PSU(p,q)) \to \mathbb{R}$, which is constant on each connected component and such that $ \vert c(\rho )\vert \leq 4p\pi (g - 1)$ for every representation $ \rho$. We show that representations with maximal characteristic number (plus some nondegeneracy condition if $ p > 2$ leave invariant a totally geodesic subspace of $ {D_{p,q}}$ isometric to $ {D_{p,p}}$.


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  • [C] Kevin Corlette, Rigid representations of Kählerian fundamental groups, J. Differential Geom. 33 (1991), no. 1, 239–252. MR 1085142
  • [DT] Antun Domic and Domingo Toledo, The Gromov norm of the Kaehler class of symmetric domains, Math. Ann. 276 (1987), no. 3, 425–432. MR 875338, 10.1007/BF01450839
  • [Go1] W. Goldman, Discontinuous groups and the Euler class, Doctoral Dissertation, University of California, Berkeley.
  • [Go2] William M. Goldman, Representations of fundamental groups of surfaces, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 95–117. MR 827264, 10.1007/BFb0075218
  • [Gr] Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
  • [K] Hellmuth Kneser, Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächenabbildungen, Math. Ann. 103 (1930), no. 1, 347–358 (German). MR 1512626, 10.1007/BF01455699
  • [M] John Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215–223. MR 0095518
  • [NS] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 0184252
  • [P] I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Translated from the Russian. Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0252690
  • [Th] W. Thurston, The geometry and topology of $ 3$-manifolds, Princeton University notes.
  • [To] Domingo Toledo, Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989), no. 1, 125–133. MR 978081
  • [Wd] John W. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257–273. MR 0293655
  • [Wf] Joseph A. Wolf, Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, pp. 271–357. Pure and App. Math., Vol. 8. MR 0404716
  • [Z] R. J. Zimmer, Ergodic theory and semi-simple groups, Birkhäuser, Boston, Mass., 1984.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1033234-5
Article copyright: © Copyright 1991 American Mathematical Society