Compact connected components in relative character varieties of punctured spheres

Compact connected components in relative character varieties of punctured spheres

Blog Article

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Apparel Hermitian Lie groups $mathrm{SU}(p,q)$ admit compact connected components.The representations in these components have several counter-intuitive properties.For instance, the image of any simple closed curve is an elliptic element.These results extend a recent work of Deroin and the first author, which treated the case of $ extrm{PU}(1,1) = mathrm{PSL}(2,mathbb{R})$.Our proof relies on the read more non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles.

The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.