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

de la Harpe, Pierre; Weber, Claude

On malnormal peripheral subgroups of the fundamental group of a

3

-manifold

de la Harpe, Pierre ; Weber, Claude

Confluentes Mathematici, Tome 6 (2014), p. 41-64 / Harvested from Numdam

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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.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/cml.12
Classification: 57M25, 57N10

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     author = {de la Harpe, Pierre and Weber, Claude},
     title = {On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold},
     journal = {Confluentes Mathematici},
     volume = {6},
     year = {2014},
     pages = {41-64},
     doi = {10.5802/cml.12},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CML_2014__6_1_41_0}
}

de la Harpe, Pierre; Weber, Claude. On malnormal peripheral subgroups  of the fundamental group of a $3$-manifold. Confluentes Mathematici, Tome 6 (2014) pp. 41-64. doi : 10.5802/cml.12. http://gdmltest.u-ga.fr/item/CML_2014__6_1_41_0/

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