Hyperbolic Jigsaws and Families of Pseudomodular Groups II.

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. [ABSTRACT FROM AUTHOR]