  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.
