Hyperbolic Jigsaws and Families of Pseudomodular Groups II

Abstract

Abstract In our previous paper [ 5], we introduced a hyperbolic jigsaw construction and constructed infinitely many noncommensurable, nonuniform, non-arithmetic lattices of $\textrm{PSL}(2,{\mathbb R})$ with cusp set $\mathbb{Q} \cup \{\infty \}$ (called pseudomodular groups by Long and Reid [ 4]), thus answering a question posed by Long and Reid. In this paper, we continue with our study of these jigsaw groups exploring questions of arithmeticity, pseudomodularity, and also related pseudo-Euclidean and continued fraction algorithms arising from these groups. We also answer another question of Long and Reid [ 4] by demonstrating a recursive formula for the tessellation of the hyperbolic plane arising from Weierstrass groups, which generalizes the well-known “Farey addition” used to generate the Farey tessellation.