Classifying torsion-free subgroups of the Picard group

Abstract

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 Soli tar 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.

Arithmetic Knots in Closed 3-Manifolds

“…For the case d -1, we deduce from the above remarks that [PSL(2,Oi) : Y] -12. However, the minimal index torsion free subgroups in PSL(2, 0\) all have 2 cusps, see [7] or [13].…”

“…Example 2: The Poincare homology sphere E contains a 2-component arithmetic link (see [7], Example D), so A(E) < 2. This prompts:…”

“…• Remark: 2ra + l is not best possible. For example the connect sum of any two Lens Spaces has a surgery description on the Borromean rings, and this link is arithmetic (see [7]). Thus the arithmetic number of such a manifold is at most 3.…”

Arithmeticity of Knot Complements

“…Proof Any maximal order 0 of M 2 [23,Chapter 3]). We shall assume in what follows that / = {a, b) for some a and b inO d .…”

“…We have seen that if K is arithmetic then F K is conjugate to a subgroup of PSL 2 (O d ) for d = 1 or 3. Since every parabolic element of PSL 2 I in F^ is F^. Consequently, F^ n F(// 2 ) = F^, so that F^ c F(// 2 ).…”