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.
| Subjects: | Geometric Topology (math.GT) |
| Cite as: | arXiv:2605.24943 [math.GT] |
| (or arXiv:2605.24943v1 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24943 arXiv-issued DOI via DataCite (pending registration) |
From: Vladimir Markovic [view email]
[v1]
Sun, 24 May 2026 08:43:25 UTC (29 KB)
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