  Let $e$ denote the Euler class on the space $Hom(\Gamma_g, PSL(2,\mathbb R))$
of representations of the fundamental group $\Gamma_g$ of the closed surface
$\Sigma_g$ of genus $g$. Goldman showed that the connected components of
$Hom(\Gamma_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$,
for $2-2g\leq k\leq 2g-2$, and that the components of Euler class $2-2g$ and
$2g-2$ consist of the injective representations whose image is a discrete
subgroup of $PSL(2,\mathbb R)$. We prove that non-faithful representations are
dense in all the other components. We show that the image of a discrete
representation essentially determines its Euler class. Moreover, we show that
for every genus and possible corresponding Euler class, there exist discrete
representations.
