# CONFLUENTES MATHEMATICI

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

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DOI : https://doi.org/10.5802/cml.12
Classification : 57M25,  57N10
Mots clés : knot, knot group, peripheral subgroup, torus knot, cable knot, composite knot, malnormal subgroup, $3$-manifold.
@article{CML_2014__6_1_41_0,
author = {Pierre de la Harpe and Claude Weber},
title = {On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold},
journal = {Confluentes Mathematici},
pages = {41--68},
publisher = {Institut Camille Jordan},
volume = {6},
number = {1},
year = {2014},
doi = {10.5802/cml.12},
mrnumber = {3266884},
zbl = {1319.57010},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.12/}
}
Pierre de la Harpe; Claude Weber. 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. https://cml.centre-mersenne.org/articles/10.5802/cml.12/

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