Genericity of hyperbolic 3-manifolds via Dehn surgery

Renxing Wan; Yanqing Zou

Genericity of hyperbolic 3-manifolds via Dehn surgery

Geometric Topology 2026-07-13 v1

Authors: Renxing Wan , Yanqing Zou

Abstract

A significant result by Lickorish and Wallace shows that every closed, orientable 3-manifold can be obtained from a Dehn surgery on one link in 3-sphere. As links and Dehn surgeries vary vastly in the universe, a question arises: how can we describe their properties in vague? We introduce a counting model on links and Dehn surgeries, and prove that under this model, (1) a randon link is hyperbolic; (2) a random 3-manifold is hyperbolic.

Keywords

hyperbolic 3-manifolds Dehn surgery random links genericity counting model

Cite

@article{arxiv.2607.11057,
  title  = {Genericity of hyperbolic 3-manifolds via Dehn surgery},
  author = {Renxing Wan and Yanqing Zou},
  journal= {arXiv preprint arXiv:2607.11057},
  year   = {2026}
}

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12 pages