Classifying torsion-free subgroups of the Picard group

Brunner, Andrew M.; Frame, Michael L.; Lee, Youn W.; Wielenberg, Norbert J.

doi:10.1090/S0002-9947-1984-0728710-2

Classifying torsion-free subgroups of the Picard group
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by Andrew M. Brunner, Michael L. Frame, Youn W. Lee and Norbert J. Wielenberg PDF

Trans. Amer. Math. Soc. 282 (1984), 205-235 Request permission

Abstract:

Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic $3$-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass, Pietrowski and Solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.

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