Lattice Subgroups of Normal Subgroups of Genus Zero of the Modular Group

1972 ◽  
Vol s3-24 (3) ◽  
pp. 449-469 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Modular Group ◽  
Normal Subgroups ◽  
Genus Zero ◽  
Lattice Subgroups

scholarly journals Some maximal normal subgroups of the modular group

1987 ◽  
Vol 30 (1) ◽  
pp. 97-101
Author(s):  
Gareth A. Jones
Keyword(s):  
Finite Group ◽  
Quotient Group ◽  
Modular Group ◽  
Galois Field ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Fractional Group

For each finite group G, let G denote the set of all normal subgroups of the modular group Γ = PSL2(ℤ) with quotient group isomorphic to G; since Γ is finitely generated, the number NG = |G| of such subgroups is finite. We shall be mainly concerned with the case where G is the linear fractional group PSL2(q) over the Galois field GF(q), in which case we shall write (q) and N(q) for G and NG; for q>3, PSL2(q) is simple, so the elements of (q) will be maximal normal subgroups of Γ.

scholarly journals Existence conditions for a class of modular subgroups of genus zero

2002 ◽  
Vol 66 (3) ◽  
pp. 517-525
Author(s):  
Joachim A. Hempel
Keyword(s):  
Rational Function ◽  
Finite Index ◽  
Modular Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Genus Zero ◽  
Existence Conditions ◽  
Necessary And Sufficient

Every subgroup of finite index of the modular groupPSL(2, ℤ) has asignatureconsisting of conjugacy-invariant integer parameters satisfying certain conditions. In the case of genus zero, these parameters also constitute a prescription for the degree and the orders of the poles of a rational functionFwith the property:Functions correspond to subgroups, and we use this to establish necessary and sufficient conditions for existence of subgroups with a certain subclass of allowable signatures.

Anamalous normal subgroups of the modular group

1983 ◽  
Vol 11 (22) ◽  
pp. 2555-2573 ◽  
Author(s):  
A.W. Mason
Keyword(s):  
Modular Group ◽  
Normal Subgroups

scholarly journals Perfect Non-Extremal Riemann Surfaces

2000 ◽  
Vol 43 (1) ◽  
pp. 115-125 ◽  
Author(s):  
Paul Schmutz Schaller
Keyword(s):  
Riemann Surface ◽  
Quadratic Forms ◽  
Riemann Surfaces ◽  
Closed Geodesic ◽  
Modular Group ◽  
Infinite Family ◽  
Positive Definite ◽  
Normal Subgroups

AbstractAn infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group PSL(2, ℤ) of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface M the length of a systole, which is a shortest closed geodesic of M.

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