Representations of the fundamental group of a closed oriented surface in Sp(4,R)

Abstract

We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus $\ge 2$ in $Sp(4,\mathbf{R})$. We prove that this is equivalent to classifying the connected components of the moduli space of representations (which consists of conjugacy classes of reductive representations). We then continue the analysis initiated by Gothen in math/9904114, relying on: (1) the relation between our moduli space of representations and the moduli space of certain (polystable) analogues of Higgs bundles (which follows essentially from work of Hitchin, Simpson and Corlette) and (2) the existence of a proper function on the moduli space whose set of local minima can be identified with the moduli space $N$ of (polystable) quadratic bundles of rank 2; we finally deduce our result from the classification of the connected components of $N$.

24 pages, submitted

Cited by in corpus (7)

• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Higgs bundles and surface group representations in the real symplectic group
• Topological Invariants of Anosov Representations
• Topology of character varieties of Abelian groups
• Deformations of maximal representations in Sp(4,R)
• Homotopy groups of moduli spaces of representations
• Compact connected components in relative character varieties of punctured spheres