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.

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We use a Koszul-type resolution to prove a weak version of Bott's vanishing theorem for smooth hypersurfaces in P n and use this result to prove a vanishing theorem for Hodge ideals associated to an effective Cartier divisor on a hypersurface. This extends an earlier result of Mustat ¸ȃ and Popa.

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.