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

de la Harpe, Pierre; Weber, Claude

doi:10.5802/cml.12

On malnormal peripheral subgroups of the fundamental group of a

3

-manifold

de la Harpe, Pierre ¹ ; Weber, Claude ¹

¹ Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse

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

Résumé

Let $K$ be a non-trivial knot in the $3$ -sphere, $E_{K}$ its exterior, $G_{K} = π_{1} (E_{K})$ its group, and $P_{K} = π_{1} (\partial E_{K}) \subset G_{K}$ its peripheral subgroup. We show that $P_{K}$ is malnormal in $G_{K}$ , namely that $g P_{K} g^{- 1} \cap P_{K} = {e}$ 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 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.

Reçu le : 2013-01-18
Révisé le : 2014-12-02
Accepté le : 2014-12-06
Publié le : 2014-09-09

MR Zbl

DOI : 10.5802/cml.12

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

Affiliations des auteurs :

de la Harpe, Pierre ¹ ; Weber, Claude ¹

¹ Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse

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de la Harpe, Pierre; Weber, Claude. On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold. Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 41-68. doi : 10.5802/cml.12. http://www.numdam.org/articles/10.5802/cml.12/

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