  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.
