Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

Arithmetic Fuchsian Groups of Genus Zero

If Γ is a finite co-area Fuchsian group acting on H, then the quotient H2/Γ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a finite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and interesting subclass of finite co-area Fuchsia...

Kronecker's Limit Formula, Holomorphic Modular Functions, and q-Expansions on Certain Arithmetic Groups

For any square-free integer N such that the “moonshine group” Γ0(N) has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of Γ0(N) to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In this article, we study the holomorphic function ...

Some Holomorphic Semi-groups.

W ∗ k = {x ǫX|xǫ Domain (A(a1) . . .A (ak)) for all a1, . . . , akǫA} is dense inX in theweak* topology. Let {e1, . . . , en} be a basis forA and setAi = A(ei), A ∗ i = A(ei). If α = (α1, . . . , αk), 1 ≤ αi ≤ n, is a sequence of integers we denote a product such as Aα1 . . .Aαk byAα; |α| = k is the length ofα. The operatorB ◦ = Σ|α|≤maαAα (the coefficients are complex numbers) is defined on Wm...

On the Structure of Some Deficiency Zero Groups

Some combinatorial aspects of finite Hamiltonian groups

In this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite Hamiltonian group. The coverings with three proper subgroups and the principal series of such a group are also counted. Finally, we give a complete description of the lattice of characteristic subgroups of a finite H...