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Topology of representation spaces of surface groups in $\mathrm{PSL}_2 (\mathbb{R})$ with assigned boundary monodromy and nonzero Euler number

Gabriele Mondello

2016
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Pure and Applied Mathematics Quarterly
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In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL 2 (R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspondence between flat bundles and Higgs bundles in the parabolic case. This extends previous results by Boden-Yokogawa and Nasatyr-Steer. A relevant portion of the paper is intended
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... to give an overview of the subject. Compared to Narasimhan-Seshadri's, the correspondence is less intuitive, since the holomorphic structure on E does not agree with the underlying holomorphic structure on the flat complex vector bundle V → S determined by the representation ρ (coming from the fact that locally constant functions are holomorphic) but it is twisted: the exact amount of such twisting is determined by the aid of the harmonic metric on V , whose existence was shown by Donaldson [16]. For G = GL N or G = SL N , the existence of the harmonic metric was proven (on any compact manifold) by Corlette [12] (and later Labourie [38] ) and the correspondence (in any dimension) was proven by Simpson [57] , who also clarified the general picture by showing [59] [60] that the fundamental objects to consider are local systems (classified by a "Betti" moduli space), vector bundles with a flat connection (classified by a "de Rham" moduli space) and holomorphic Higgs bundles (classified by a "Dolbeault" moduli space) and by constructing their moduli spaces. Correspondence for SL 2 (R). Back to the rank 2 case, among the many results contained in [31], Hitchin could determine which Higgs bundles correspond to monodromies of hyperbolic metrics, thus parametrizing Teichmüller space by holomorphic quadratic differentials on (S, I) (note that the Higgs field in Hitchin's work identifies to the Hopf differential of the harmonic map in Wolf's parametrization [68] ). Moreover, the space of isomorphism classes of Higgs bundles (E, Φ) carries a natural S 1 -action u · (E, Φ) = (E, uΦ), which is also rather ubiquitous when dealing with harmonic maps with a two-dimensional domain (for instance [8] ); in rank 2, the locus fixed by the (−1)-involution [(E, Φ)] ↔ [(E, −Φ)] is identified to the locus of unitary (if Φ = 0) or real (if Φ = 0) representations. This allows Hitchin to fully determine the topology of the connected components of Rep(S, PSL 2 (R)) with non-zero Euler number as that of a complex vector bundle over a symmetric product of copies of S. The real component with Euler number zero seems slightly subtler to treat, since it contains certain classes of reducible representations (or, equivalently, of strictly semi-stable Higgs bundles) for which the correspondence does not hold. Surfaces with punctures Let S be a compact connected oriented surface and let P = {p 1 , . . . , p n } ⊂ S be a subset of n distinct marked points. Denote byṠ the punctured surface S \ P and assume χ(Ṡ) < 0.

doi:10.4310/pamq.2016.v12.n3.a3
fatcat:ys6a5fbyj5gkhaf2lu22hcl4ka