Quasi-Fuchsian vs Negative curvature metrics on surface groups

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 $\Teich(Σ)$ is the only common part of these two families, even when viewed from the coarse-geometric perspective.

12 pages