Non-injective representations of a closed surface group into $PSL(2,\mathbb R)$

Details Non-injective representations of a closed surface group into $PSL(2,\mathbb R)$ Non-injective representations of a closed surface group into $PSL(2,\mathbb R)$

2005-02-28

arxiv.org/abs/math/0502585v1

Mathematics

Geometric Topology

15 pages, 2 figures

Scientific paper

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.

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