On malnormal peripheral subgroups of the fundamental group of a 3-manifold
Confluentes Mathematici, Volume 6 (2014) no. 1, pp. 41-68.

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 (E K )G K its peripheral subgroup. We show that P K is malnormal in G K , namely that gP K g -1 P K ={e} for any gG K with gP 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.

Received: 2013-01-18
Revised: 2014-12-02
Accepted: 2014-12-06
Published online: 2014-09-09
Classification: 57M25,  57N10
Keywords: knot, knot group, peripheral subgroup, torus knot, cable knot, composite knot, malnormal subgroup, 3-manifold.
     author = {Pierre de la Harpe and Claude Weber},
     title = {On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {6},
     number = {1},
     year = {2014},
     pages = {41-68},
     language = {en},
de la Harpe, Pierre; Weber, Claude. On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold. Confluentes Mathematici, Volume 6 (2014) no. 1, pp. 41-68. https://cml.centre-mersenne.org/item/CML_2014__6_1_41_0/

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