A discreteness algorithm for 4-punctured sphere groups

Caleb Ashley

A discreteness algorithm for 4-punctured sphere groups

Geometric Topology 2020-12-02 v2

Authors: Caleb Ashley

Abstract

Let $\Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $\Gamma$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing related works and future directions of inquiry.

Keywords

sphere packings and spherical designs wavelet analysis gamma-convergence and homogenization

Cite

@article{arxiv.1910.10055,
  title  = {A discreteness algorithm for 4-punctured sphere groups},
  author = {Caleb Ashley},
  journal= {arXiv preprint arXiv:1910.10055},
  year   = {2020}
}

Comments

20 pages, 2 figures, to appear in "The Golden Anniversary Celebration of the National Association of Mathematicians," published by AMS Contemporary Mathematics. Volume: 759; 2020. Print ISBN: 978-1-4704-5130-1

A discreteness algorithm for 4-punctured sphere groups

Abstract

Keywords

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