Action of the Mapping Class Group on Character Varieties and Higgs Bundles

Graeme Wilkin ; Oscar Garcia-Prada

Documenta mathematica, Tome 25 (2020), pp. 841-868

Résumé

We consider the action of a finite subgroup of the mapping class group Mod(S) of an oriented compact surface S of genus g⩾2 on the moduli space R(S,G) of representations of π1​(S) in a connected semisimple real Lie group G. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element J in the Teichmüller space of S for which Γ can be realised as a subgroup of the group of automorphisms of X=(S,J) which are holomorphic or antiholomorphic. We identify the fixed points of the action of Γ on R(S,G) in terms of G-Higgs bundles on X equipped with a certain twisted Γ-equivariant structure, where the twisting involves abelian and non-abelian group cohomology simultaneously. These, in turn, correspond to certain representations of the orbifold fundamental group. When the kernel of the isotropy representation of the maximal compact subgroup of G is trivial, the fixed points can be described in terms of familiar objects on Y=X/Γ+, where Γ+⊂Γ is the maximal subgroup of Γ consisting of holomorphic automorphisms of X. If Γ=Γ+ one obtains actual Γ-equivariant G-Higgs bundles on X, which in turn correspond with parabolic Higgs bundles on Y=X/Γ (this generalizes work of Nasatyr Steer for G=SL(2,R) and Boden, Andersen Grove and Furuta Steer for G=SU(n)). If on the other hand Γ has antiholomorphic automorphisms, the objects on Y=X/Γ+ correspond with pseudoreal parabolic Higgs bundles. This is a generalization in the parabolic setup of the pseudoreal Higgs bundles studied by the first author in collaboration with Biswas Hurtubise.

Détail

DOI : 10.4171/dm/764

Classification : 14H60, 58D27, 58D29
Mots-clés : mapping class group, moduli space, character variety, Higgs bundle, surface group representation, parabolic Higgs bundle, equivariant structure