Nil 3-manifolds and cusps of complex hyperbolic surfaces

Julien Paupert; Connor Sell

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arXiv · 2411.15345

Nil 3-manifolds and cusps of complex hyperbolic surfaces

Julien Paupert·Connor Sell

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Abstract

McReynolds showed that every compact Nil 3-manifold occurs as the cusp cross-section of some arithmetic complex hyperbolic 2-manifold. We classify which commensurability classes of cusped, arithmetic, complex hyperbolic 2-manifolds admit cusps with cross-section homeomorphic to a given compact Nil 3-manifold. In particular, there are some Nil 3-manifolds which occur as cusps in every such commensurability class, and some which only occur in a single commensurability class. We also show that every compact Nil 3-manifold occurs as the cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold.

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Julien Paupert, Connor Sell. 2024-11-22. Nil 3-manifolds and cusps of complex hyperbolic surfaces. https://arxiv.org/abs/2411.15345

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