On malnormal peripheral subgroups in fundamental groups of 3-manifolds

Abstract

Let $K$ be a non-trivial knot in the 3-sphere, $E_K$ its exterior, $G_K = \pi_1(E_K)$ its group, and $P_K = \pi_1(\partial E_K) \subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \cap P_K = \{e\}$ for any $g \in G_K$ with $g \notin P_K$, 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.