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 $\text{genus}(\Sigma )\ge 2$, and $\Gamma \subset \mathrm{S}L(2,\mathbb{C})$ is a cocompact lattice, such that there is a generically injective holomorphic map $\Sigma ⟶\mathrm{S}L(2,\mathbb{C})/\Gamma$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.

33 pages; 3 figures; comments are very welcome