Holomorphic curves in compact quotients of SL(2,C)
Geometric Topology
2026-05-26 v1
Abstract
We prove that every discrete faithful representation of the surfcae group into SL(2,C) is the monodromy of a holomorphic connection on the trivial rank-2 vector bundle over a Riemann surface. As an application, we answer the question posed by Ghys and Huckleberry-Winkelmann (known as the Margulis' problem) by proving that every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two. The main tools we use are the Non-Abelian Hodge correspondence, the WKB analysis, and the Morgan-Shalen compactification.
Cite
@article{arxiv.2605.24943,
title = {Holomorphic curves in compact quotients of SL(2,C)},
author = {Yiran Lin and Vladimir Markovic},
journal= {arXiv preprint arXiv:2605.24943},
year = {2026}
}