Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where $\widetilde\Sigma\subset\mathbb{H}^3$ is the universal covering of $\Sigma$, and provided $\Gamma\leq\Gamma_{0}\in$$PSL(2,\mathbb{C})$.

My question is that if $\widetilde\Sigma$ is an embedding in $\mathbb{H}^3$, or it is just an immersion? Or what conditions on $\Sigma$ guarantee that $\widetilde\Sigma\subset\mathbb{H}^3$ is an embedding? (Obviously, it is an embedding when $\Sigma$ is embedded in $M$. )

Thanks for answering.