Higgs bundles over cell complexes and representations of finitely presented groups

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JournalPacific Journal of Mathematics
DateAccepted/In press - 23 Feb 2018
DatePublished (current) - 1 May 2018
Issue number1
Number of pages25
Pages (from-to)31-55
Original languageEnglish


The purpose of this paper is to extend the Donaldson–Corlette theorem to the case of vector bundles over cell complexes. We define the notions of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the SL(r,C) character variety of a finitely presented group Γ is homeomorphic to the moduli space of rank-r Higgs bundles over an admissible complex X with π1(X)=Γ. A key role is played by the theory of harmonic maps defined on singular domains.

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