New Examples from the Jigsaw Groups Construction

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.