Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Abstract

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant.

v2: added due credits to the work of Burger, Iozzi and Wienhard. v3: corrected count of connected components for G=SU(p,q) (p \neq q); added due credits to the work of Xia and Markman-Xia; minor corrections and clarifications. 31 pages

References in corpus (8)

• Surface group representations in PU(p,q) and Higgs bundles
• Bounded Cohomology and Geometry
• Representations of the fundamental group of a surface in PU(p,q) and holomorphic triples
• Representations of the fundamental group of a closed oriented surface in Sp(4,R)
• The Moduli of Flat PU(p,p)-Structures with Large Toledo Invariants
• Maximal Representations of Surface Groups: Symplectic Anosov Structures
• Surface group representations with maximal Toledo invariant
• Anosov Flows, Surface Groups and Curves in Projective Space

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