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

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

DOI: https://doi.org/10.1090/S0002-9947-1984-0728710-2

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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.

References

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

The subgroups of a polygonal product of groups

Combinatorial group theory

Robert Riley, Discrete parabolic representations of link groups, Mathematika 22 (1975), no. 2, 141–150. MR 425946, DOI 10.1112/S0025579300005982
R. Riley, Seven excellent knots, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 81–151. MR 662430
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 284516, DOI 10.1016/0001-8708(71)90027-2
W. Thurston, Hyperbolic geometry and $3$-manifolds, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 9–25. MR 662424
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Norbert J. Wielenberg, Hyperbolic $3$-manifolds which share a fundamental polyhedron, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 505–513. MR 624835