Projects per year
Abstract
We consider the action of a finite subgroup of the mapping class group $\Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\calR(S,G)$ of representations of $\pi_1(S)$ 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 Teichm\"uller space of $S$ for which $\Gamma$ 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 $\Gamma$ on $\calR(S,G)$ in terms of $G$Higgs bundles on $X$ equipped with a certain twisted $\Gamma$equivariant structure, where the twisting involves abelian and nonabelian 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 $Y=X/\Gamma^+$, where $\Gamma^+ \subset \Gamma$ is the maximal subgroup of $\Gamma$ consisting of holomorphic automorphisms of $X$. If $\Gamma=\Gamma^+$ one obtains actual $\Gamma$equivariant $G$Higgs bundles on $X$, which in turn correspond with parabolic Higgs bundles on $Y=X/\Gamma$ (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 $\Gamma$ has antiholomorphic automorphisms, the objects on $Y=X/\Gamma^+$ 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.
Original language  English 

Number of pages  28 
Journal  Documenta Mathematica 
Publication status  Accepted/In press  22 Jun 2020 
Bibliographical note
This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy. Further copying may not be permitted; contact the publisher for details.Projects
 1 Finished

Geometry and Topology of Singular Spaces
1/08/18 → 30/06/19
Project: Other project › Project from former institution