Surface group representations to $SL(2,{\mathbb C})$ and Higgs bundles with smooth spectral data

TL;DR

This paper demonstrates that for any non-elementary surface group representation into SL(2,C), one can find a Riemann surface structure making the associated Higgs bundle lie outside the Hitchin discriminant locus, revealing new geometric insights.

Contribution

It establishes the existence of Riemann surface structures for which the Higgs bundle associated to a non-elementary representation is outside the Hitchin discriminant locus, advancing understanding of Higgs bundle moduli.

Findings

Existence of Riemann surfaces with desired Higgs bundle properties

Non-elementary representations can be realized with smooth spectral data

Higgs bundles can be positioned outside the discriminant locus

Abstract

We show that for every nonelementary representation of a surface group into $SL(2,{\mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.