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Congruence Subgroups of the Picard Group

The Picard group Γ = PSL 2 (Z [i]) is the group of linear fractional transformations with ad – bc = ± 1 and a, b, c, d Gaussian integers. Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.