Higgs bundles over cell complexes and representations of finitely presented groups

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.