Hyperbolic jigsaws and families of pseudomodular groups I

TL;DR

This paper demonstrates the existence of infinitely many distinct pseudomodular groups with all rational cusps, using a new hyperbolic jigsaw construction for Fuchsian groups' fundamental domains.

Contribution

It introduces hyperbolic jigsaw groups, a novel method for constructing Fuchsian groups with specific cusp properties, answering a longstanding question.

Findings

Infinitely many pseudomodular groups exist.

Hyperbolic jigsaw groups can be constructed via gluing ideal triangles.

These groups are not commensurable to the modular group.

Abstract

We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.