Quasi-Fuchsian vs Negative curvature metrics on surface groups

TL;DR

This paper compares negative curvature metrics and quasi-Fuchsian representations on surface groups, showing that their only common geometric structure is the Teichmüller space, highlighting the distinct nature of these metric families.

Contribution

It demonstrates that the intersection of negative curvature metrics and quasi-Fuchsian representations on surface groups is precisely the Teichmüller space, even from a coarse-geometry perspective.

Findings

Teichmüller space is the only intersection of the two metric families.

Negative curvature metrics and quasi-Fuchsian representations are largely distinct.

The intersection remains the same under coarse-geometric analysis.

Abstract

We compare two families of left-invariant metrics on a surface group $\Gamma=\pi_1(\Sigma)$ in the context of course-geometry. One family comes from Riemannian metrics of negative curvature on the the surface $\Sigma$, and another from quasi-Fuchsian representations of $\Gamma$. We show that the Teichmuller space $\Teich(\Sigma)$ is the only common part of these two families, even when viewed from the coarse-geometric perspective.