The Desymmetrized PSL(2, Z) Group; Its ‘Square-Box’ One-Cusp Congruence Subgroups †

: The desymmetrized PSL(2, Z ) group is studied. The Fourier coe ﬃ cients of the non-holomorphic one-cusp Eisenstein series expansion are summed; as a new further result, a new dependence on the Euler’s γ constant is found. The congruence subgroups of the desymmetrized PSL ( 2, Z ) are scrutinized, and the related structures are investigated. A ’square-box’ one-cusp congruence subgroup is constructed. New leaky tori are constructed.


Introduction
The desymmetrized PSL(2, Z) group is considered. In [1], the motivations of the choice of the desymmetrized group are given, as it contains the parabolic element.
The well-posed-ness of the summation of the Fourier coefficients is assured after the control of the meromorphic continuability of the ζ k (s) functions, as in [3]; the further complements of the meromorphic continuability, as well as the controls on the summability of the series, from the points of view both analytical and of the number-theoretical issues, are briefly recapitulated in [4].
In the present paper, the summation of the Fourier coefficients on the non-holomorphic Eisenstein series of the one-cusp desymmetrized PSL(2, Z) group is achieved [4], and a new dependence on the Euler γ constant is found.
The congruence subgroups of the desymmetrized PSL(2, Z) group are newly constructed; new constructions of the leaky torus are presented after these constructions [9]. It is important to recall that the (sub)-grouppal structures have a role in the study of the modular Monster group [10].

Materials and Methods
In [11], the use of the divisor function in the faltungs of the Fourier coefficients in the Eisenstein-Maass series is outlined.
In [2] Ch. 11.4, the distribution of the Fourier coefficients of the non-holomorphic one-cusp form of the desymmetrized PSL(2, Z) is found as the Fourier expansion of the Eisenstein series E(z; s) where the coefficients φ(s) and φ n (s) are defined in [2] in p. 508 as and in p. 76 the following specification is spelled out where the latter term is summed for the considered grouppal structure as The non-holomorphic Eisenstein series E(z, s) therefore becomes where and a n (y, s)

Results
The Eisenstein series Equation (1) is summed after summing the Fourier coefficients Equation (4) after applying the Dirichlet formula, after which one now here thus for the terms following Equation (5) and, for s ≥ 1, γ being the Euler's γ constant. Formula Equation (4) is now summed as the dependence of the term φ m (s) in Equation (4) is now summed as with a new dependence on the Euler's γ constant.

The Congruence Subgroup Γ 0 of PSL(2, Z)
The 'square-box' congruence subgroup Γ 0 of PSL(2, Z) is constructed on the one-cusp 'square-box' domain c : x 2 + y 2 = 1, (11c) and generated after the (hyperbolic) reflections After the Markoff uniqueness property [12], there exists an isometry between any two simple closed geodesics of equal length on a torus; furthermore, the Laplace-Beltrami operator on Riemann surfaces of constant negative curvature is proven to have rigidity property. It is therefore possible to present new constructions of leaky tori.

A New Gutzwiller Leaky Tori
A leaky torus is defined in [5]; it is obtained after unfolding the PSL(2, Z) [6] according to the triangular domains of the PSL(2, Z) domain in a congruence subgroup of PSL(2, Z) domain.
The leaky torus in [8] is equivalent to that of [6] p. 181 with respect to both the domain and the generators.
A new Gutzwiller leaky torus is here generated after the reflections and on the domain of sides C n , defined as in the limit n → ∞.

A New Leaky Torus from the Desymmetrized Triangle Group
It is possible to construct a leaky torus from the desymmetrized domain of the PSL(2, Z) group. The leaky torus is thus constructed after the unfolding of the chosen trajectory according to the domain of the desymmetrized (triangular) PSL(2, Z) group; the leaky torus is generated after the generators on the domain delimited after the sides C n defined as

A New Leaky Torus from the 'Square-Box' Congruence Subgroup
A new leaky torus from the congruence subgroups of Γ 0 of PSL(2, Z) is obtained here after unfolding [13,14] the 'square box' of the Γ 0 congruence subgroup of the desymmetrized PSL(2, Z) group domain into the congruence subgroup Γ 0 (N) in the limit N → ∞. This new leaky torus is defined on the domain of sides c n , constructed as and generated after the reflections T 1,n+1 : z → z = T −1 n+1 R 1 T n+1 z, n + 1 2 < x ≤ n + 1, T 1,n+1 : z → z = T −1 n+1 R 2 T n+1 z, n < x ≤ n + 1 2 (20b) (which contain, of course, also the case R 1 , R 2 ).
The new generators Equation (20) identify arcs of circumferences that are the sides of two different 'square boxes' congruence subgroups delimiting the domain of the congruence subgroups of the desymmetrized PSL(2, Z) group.