Holomorphic curves in compact quotients of SL(2,C)

Yiran Lin; Vladimir Markovic

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arXiv · 2605.24943

Holomorphic curves in compact quotients of SL(2,C)

Yiran Lin·Vladimir Markovic

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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.

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BibTeX RIS

Yiran Lin, Vladimir Markovic. 2026-05-24. Holomorphic curves in compact quotients of SL(2,C). https://arxiv.org/abs/2605.24943

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