Quasi-Fuchsian vs negative curvature metrics on surface groups

Abstract

We compare two families of left-invariant metrics on a surface group Γ = π1(Σ) in the context of course-geometry. One family comes from Riemannian metrics of negative curvature on the surface Σ, and another from quasi-Fuchsian representations of Γ. We show that the Teichmüller space $${\cal T}$$ (Σ) is the only common part of these two families, even when viewed from the coarse-geometric perspective.