Components of spaces of representations and stable triples

Abstract

We consider the moduli spaces of representations of the fundamental group of a surface of genus g 2 in the Lie groups SU(2, 2) and Sp(4, R). It is well known that there is a characteristic number, d, of such a representation, satisfying the inequality |d| 2g − 2. This allows one to write the moduli space as a union of subspaces indexed by d, each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d = ±(2g − 2) are connected in the case of representations in SU(2, 2), while they break up into 3 · 2 2g + 2g − 4 connected components in the case of representations in Sp(4, R). We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and García-Prada.

Surface group representations with maximal Toledo invariant

“…The global properties of these components were investigated by García-Prada, Bradlow, and Gothen using Higgs bundles [29], [4], and [5] and the geometric properties of maximal representations were investigated by the authors in [13], [12].…”

Untitled

“…The cases when the structure groups being U(p, 1) and PU (2,1) are treated in [21,22]. Gothen obtained partial results for the structure groups SU (2,2) and Sp(4, R) [8]. Other related results have been obtained in [19,20].…”

Representations of the fundamental group of a surface in and holomorphic triples

“…There is a link between the stability conditions for holomorphic triples and U(p, q)-Higgs bundles: one can show (see [8]) that a U(p, q)-Higgs bundle (E, φ) with b = 0 or c = 0 is (semi-)stable if and only if the corresponding holomorphic triple (E 1 , E 2 , φ) is α -(semi-)stable for α = 2g − 2 . Lemma 2 then implies the following result.…”

“…Des résultats antérieurs concernant ce problème pour des groupes réels non compacts G sont dusà Goldman [7], Gothen [8], Hitchin [10], Markman et Xia [11], et Xia [17,18,19].…”