New Examples from the Jigsaw Groups Construction

TL;DR

This paper introduces a new infinite family of pseudomodular groups constructed via the jigsaw method, expanding understanding of their existence and properties, and also identifies some jigsaw groups that are not pseudomodular.

Contribution

It constructs a novel infinite family of pseudomodular groups using the jigsaw construction and analyzes their properties, addressing open questions about pseudomodularity.

Findings

Established a new infinite family of pseudomodular groups

Discovered that some simple jigsaw groups are not pseudomodular

Provided partial answers to existing questions about pseudomodular groups

Abstract

A pseudomodular group is a discrete subgroup $\Gamma \leq PGL(2,\mathbb{Q})$ which is not commensurable with $PSL(2,\mathbb{Z})$ and has cusp set precisely $\mathbb{Q}\cup\{\infty\}$. The existence of such groups was proved by Long and Reid. Later, Lou, Tan and Vo constructed two infinite families of non-commensurable pseudomodular groups which they called jigsaw groups. In this paper we construct a new infinite family of non-commensurable pseudomodular groups obtained via this jigsaw construction. We also find that infinitely many of the simplest jigsaw groups are not pseudomodular, providing a partial answer to questions posed by the aforementioned authors.