On the existence of holomorphic curves in compact quotients of $\mathrm{SL}(2,\mathbb C)$

Abstract

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $\Sigma \, \longrightarrow\, {\mathrm SL}(2, \mathbb C)/\Gamma$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.