The Space of Vectored Hyperbolic Surfaces is Path-Connected

Sangsan Warakkagun

The Space of Vectored Hyperbolic Surfaces is Path-Connected

Geometric Topology 2024-08-29 v2

Authors: Sangsan Warakkagun

Abstract

In the space $\mathcal{H}^2$ of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov-Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of $\rm{PSL}_2(\mathbb{R})$ . Using paths constructed from changing the Fenchel-Nielsen coordinates and shrinking simple closed curves to cusps, we demonstrate path-connectivity of $\mathcal{H}^2$ and some of its subspaces.

Keywords

hyperbolic surfaces path-connectivity gromov-hausdorff convergence chabauty topology fenchel-nielsen coordinates

Cite

@article{arxiv.2311.16667,
  title  = {The Space of Vectored Hyperbolic Surfaces is Path-Connected},
  author = {Sangsan Warakkagun},
  journal= {arXiv preprint arXiv:2311.16667},
  year   = {2024}
}

Comments

11 pages, 3 figure; substantially corrected and edited from the previous version; accepted, to appear in Conformal Geometry and Dynamics

The Space of Vectored Hyperbolic Surfaces is Path-Connected

Abstract

Keywords

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