On malnormal peripheral subgroups of the fundamental group of a $3$-manifold

de la Harpe, Pierre; Weber, Claude

Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 41-68.

Let $K$ be a non-trivial knot in the $3$-sphere, $E_K$ its exterior, $G_K={\pi }_1\left(E_K\right)$ its group, and $P_K={\pi }_1\left(\partial E_K\right)\subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $g{P}_K{g}^{-1}\cap P_K=\left\{e\right\}$ 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 of which the boundary is a non-empty union of tori (Theorem 3). Proofs are written with non-expert readers in mind. Half of our paper (Appendices A to D) is a reminder of some three-manifold topology as it flourished before the Thurston revolution.

In a companion paper [15], we collect general facts on malnormal subgroups and Frobenius groups, and we review a number of examples.

Classification : 57M25, 57N10
Mots-clés : knot, knot group, peripheral subgroup, torus knot, cable knot, composite knot, malnormal subgroup, $3$-manifold.

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