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

TL;DR

This paper proves the existence of a compact Riemann surface of genus at least 2 that admits a generically injective holomorphic map into a quotient of SL(2,C) by a cocompact lattice, answering longstanding questions.

Contribution

It establishes the existence of such holomorphic curves in compact quotients of SL(2,C), resolving questions posed by Huckleberry, Winkelmann, and Ghys.

Findings

Existence of a compact Riemann surface with genus ≥ 2 mapping into SL(2,C)/Γ

Construction of a cocompact lattice Γ in SL(2,C)

Positive answer to open questions about holomorphic curves in these quotients

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}.