On malnormal peripheral subgroups in fundamental groups of 3-manifolds

2011/04/15 by Pierre de la Harpe, de la Harpe, Pierre, Claude Weber +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1104.3062

arxiv created 2011/04/15 · arxiv updated 2011/04/18

Abstract

Let K be a non-trivial knot in the 3-sphere, E_K its exterior, G_K = π₁(E_K) its group, and P_K = π₁(∂ E_K) ⊂ G_K its peripheral subgroup. We show that P_K is malnormal in G_K, namely that gP_Kg^-1 ∩ P_K = \e\ for any g ∈ G_K with g ∉ P_K, unless K is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in E_K attached to T_K which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental group of a compact orientable irreducible 3-manifold with boundary a non-empty union of tori (Theorem 3). Proofs are written with non-expert readers in mind. Half of our paper (Sections 7 to 10) is a reminder of some three-manifold topology as it flourished before the Thurston revolution. In a companion paper [HaWeOs], we collect general facts on malnormal subgroups and Frobenius groups, and we review a number of examples.

Related

• On embedding the fundamental group of a 3-manifold in one of its knot groups
• Peripheral subgroups of Kleinian groups
• Profinite rigidity, fibering, and the figure-eight knot
• Some 3-manifolds and 3-orbifolds with large fundamental group
• Topological Invariants of 3-Manifolds with Boundary