Embedding closed totally geodesic surfaces in Bianchi orbifolds

Junehyuk Jung; Alan W. Reid

Embedding closed totally geodesic surfaces in Bianchi orbifolds

Number Theory 2020-03-12 v1 Geometric Topology

Authors: Junehyuk Jung , Alan W. Reid

Abstract

We study embedding of closed totally geodesic hyperbolic 2-orbifolds in the Bianchi orbifolds $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$ . Our main result shows that there is a constant $c$ such that for $d$ large enough there are at least $cd$ closed embedded totally geodesic hyperbolic 2-orbifolds. Moreover we provide a list which conjecturally consists of those $d$ for which $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$ does not contain a closed embedded totally geodesic hyperbolic 2-orbifold.

Keywords

hyperbolic 3-manifolds hyperbolic surfaces hyperbolic geometry

Cite

@article{arxiv.2003.05427,
  title  = {Embedding closed totally geodesic surfaces in Bianchi orbifolds},
  author = {Junehyuk Jung and Alan W. Reid},
  journal= {arXiv preprint arXiv:2003.05427},
  year   = {2020}
}

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22 pages

Embedding closed totally geodesic surfaces in Bianchi orbifolds

Abstract

Keywords

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