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Congruence Subgroups of the Picard Group

Published online by Cambridge University Press:  20 November 2018

Benjamin Fine
Benjamin Fine*
Affiliation:
Fairfield University, Fairfield, Connecticut
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The Picard group Γ = PSL2 (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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1474 - 1481
Copyright
Copyright © Canadian Mathematical Society 1980

References

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