The normal subgroup structure of the Picard group

Fine, Benjamin; Newman, Morris

doi:10.1090/S0002-9947-1987-0891646-3

The normal subgroup structure of the Picard group
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Trans. Amer. Math. Soc. 302 (1987), 769-786 Request permission

Abstract:

The Picard group $\Gamma$ is $PS{L_2}(Z[i])$, the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of $\Gamma$. In particular we give a complete classification of the normal subgroups for indices less than $60$ and show that beyond this there are large gaps in the possible indices. This classification depends on the structure of the derived series. Finally we give examples of normal noncongruence subgroups.

References

Andrew M. Brunner, Michael L. Frame, Youn W. Lee, and Norbert J. Wielenberg, Classifying torsion-free subgroups of the Picard group, Trans. Amer. Math. Soc. 282 (1984), no. 1, 205–235. MR 728710, DOI 10.1090/S0002-9947-1984-0728710-2
Andrew M. Brunner, Youn W. Lee, and Norbert J. Wielenberg, Polyhedral groups and graph amalgamation products, Topology Appl. 20 (1985), no. 3, 289–304. MR 804041, DOI 10.1016/0166-8641(85)90096-3
Benjamin Fine, Fuchsian subgroups of the Picard group, Canadian J. Math. 28 (1976), no. 3, 481–485. MR 473035, DOI 10.4153/CJM-1976-049-3
Benjamin Fine, Congruence subgroups of the Picard group, Canadian J. Math. 32 (1980), no. 6, 1474–1481. MR 604701, DOI 10.4153/CJM-1980-116-7

Fuchsian embeddings in the Bianchi groups

A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255. MR 260879, DOI 10.1090/S0002-9947-1970-0260879-9
A. Karrass and D. Solitar, Subgroups of $\textrm {HNN}$ groups and groups with one defining relation, Canadian J. Math. 23 (1971), 627–643. MR 301102, DOI 10.4153/CJM-1971-070-x

Vorlesungen uber der Theorie die Modulfunktionen

Alexander Lubotzky, Free quotients and the congruence kernel of $\textrm {SL}_{2}$, J. Algebra 77 (1982), no. 2, 411–418. MR 673125, DOI 10.1016/0021-8693(82)90263-0

Non-Euclidean tesselations and their groups

Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283

Normal congruence subgroups of the modular group

85

Morris Newman, Free subgroups and normal subgroups of the modular group, Illinois J. Math. 8 (1964), 262–265. MR 159865
Morris Newman, A complete description of the normal subgroups of genus one of the modular group, Amer. J. Math. 86 (1964), 17–24. MR 163966, DOI 10.2307/2373033
Mutuo Takahasi, Note on word-subgroups in free products of groups, J. Inst. Polytech. Osaka City Univ. Ser. A 2 (1951), 13–18. MR 45723