  In this paper we prove that every open Riemann surface properly embeds in the
Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve,
where $SL_2(\mathbb{C})$ is endowed with its standard contact structure.
  As a consequence, we derive the existence of proper, weakly complete, flat
fronts in the real hyperbolic space $\mathbb{H}^3$ with arbitrary complex
structure.
