Classifying Torsion-Free Subgroups of the Picard Group
Copyright policy )Classifying Torsion-Free Subgroups of the Picard Group
Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic 3 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.
discrete subgroup of PSL(2,\({\mathbb{C}})\), Topology of general \(3\)-manifolds, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, complement of a link in a closed 3-manifold, polygonal product, hyperbolic 3-space, finite hyperbolic volume, Subgroup theorems; subgroup growth, Discrete subgroups of Lie groups, torsion- free subgroups, generalized free product, surgery, projective special linear group, Knots and links in the \(3\)-sphere, Mappings of semigroups, Picard group, hyperbolic 3-manifold
discrete subgroup of PSL(2,\({\mathbb{C}})\), Topology of general \(3\)-manifolds, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, complement of a link in a closed 3-manifold, polygonal product, hyperbolic 3-space, finite hyperbolic volume, Subgroup theorems; subgroup growth, Discrete subgroups of Lie groups, torsion- free subgroups, generalized free product, surgery, projective special linear group, Knots and links in the \(3\)-sphere, Mappings of semigroups, Picard group, hyperbolic 3-manifold
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