Simple loop conjecture for discrete representations in PSL$(2,\mathbb{R})$

We show that the Simple Loop Conjecture holds for any representation $\rho \text{\hspace{0.17em}⁣}:{\pi }_1(S)⟶\text{PSL}(2,\mathbb{R})$ that is discrete but not faithful. That is, we show the existence of a simple closed curve in the kernel of such a representation.