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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 discretesubgroup of $PSL(2,\mathbb R)$. We prove that non-faithful representations aredense in all the other components. We show that the image of a discreterepresentation essentially determines its Euler class. Moreover, we show thatfor every genus and possible corresponding Euler class, there exist discreterepresentations.Comment: 15 pages, 2 figure

Non-injective representations of a closed surface group into PSL(2, )