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Transactions of the American Mathematical Society

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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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Article copyright: © Copyright 1991 American Mathematical Society