The desymmetrized PSL(2, Z) group; its 'square-box' one-cusp congruence subgroups

The desymmetrized PSL(2, Z) group and its 'square-box' one-cusp congruence subgroups
The desymmetrized PSL(2, Z) group is considered.
The Fourier coefficients of the non-holomorphic one-cusp Eisenstein series of the PSL(2, Z) group are summed ; for this purpose, the
well-posedness (i.e. the meromorphic continuability of the pertinent objects) is controlled. As a further result, a new
dependence on the Euler's gamma constant is also found.
The structures related to the the desymmetrized PSL(2, Z) congruence subgroups are investigated. As a first instance,
the 'square-box' one-cusp congruence subgroup is constructed.
As a second instance, the commutator subgroups of the 'square-box' one-cusp congruence subgroups of the desymmetrized
PSL(2, Z) group, endowed with two (hyperbolic) reflections, are exaxmined. The corresponding leaky tori are defined.
The possible relations with the modular Monster group are envisaged.