THE DESYMMETRIZED PSL(2,Z) GROUP, THE ‘SQUARE BOX’ CONGRUENCE SUBGROUP, THE CONGRUENCE SUBGROUPS AND LEAKY TORI

Abstract

The desymmetrized PSL(2,Z) group is considered. The congruence subgroups of the desymmetrized PSL(2,Z) group are analysed according to the hyperbolic reflections contained. The Laplace-Beltrami problem and the Hamiltonian problem associated with the Laplace-Beltrami operator are discussed according to the boundary conditions for the grouppal structures. New constructions of the leaky tori are found; in particular, they are obtained after the generators of the PSL(2,Z) group, from the desymmetrized-triangle PSL(2,Z) group, and from the ’square-box’ Γ0 congruence subgroup of the desymmetrized PSL(2,Z) group