Surface groups are frequently faithful

Abstract

We show that the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the ${\mathrm{PSL}}_2(\mathbb{K})$ representation variety, where $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$, answering a question of W. M. Goldman. We also prove the existence of faithful representations into $\mathrm{PU}(2,1)$ with certain nonintegral Toledo invariants.