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

Abstract

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 $2-2g\le k\le 2g-2$, 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.

15 pages, 2 figures

Cited by in corpus (1)

• Connected components of the compactification of representation spaces of surface groups