Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

TL;DR

This paper introduces algorithms to determine discreteness, solve the constructive membership problem, and compute fundamental domains for finitely generated subgroups of SL_2(R), with implementations in Magma for algebraic number fields.

Contribution

It provides the first practical algorithms for recognizing and constructing fundamental domains of discrete subgroups of SL_2(R).

Findings

Algorithms successfully determine discreteness.

Constructive membership problem is solved.

Fundamental domains can be computed effectively.

Abstract

We provide algorithms to decide whether a finitely generated subgroup of $\mathrm{SL}_2(\mathbb{R})$ is discrete, solve the constructive membership problem for finitely generated discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$, and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.