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Pierre de la Harpe; Claude Weber
On malnormal peripheral subgroups of the fundamental group of a $3$-manifold
Confluentes Mathematici, 6 no. 1 (2014), p. 41-68, doi: 10.5802/cml.12
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Class. Math.: 57M25, 57N10
Keywords: knot, knot group, peripheral subgroup, torus knot, cable knot, composite knot, malnormal subgroup, $3$-manifold.

Résumé - 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 = \lbrace e\rbrace $ 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.

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