A discreteness algorithm for 4-punctured sphere groups

TL;DR

This paper introduces an algorithm to determine the discreteness of groups generated by three parabolic transformations in PSL(2,R), connecting various mathematical paradigms and providing foundational insights for hyperbolic surface structures.

Contribution

It presents a novel algorithm for testing discreteness of 4-punctured sphere groups and clarifies the equivalence of different mathematical approaches to this problem.

Findings

Algorithm effectively determines discreteness of the group

Unifies multiple mathematical paradigms for discreteness

Provides foundational exposition for future research

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.