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 =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.

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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.
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/

[1] R.H. Bing and Joseph M. Martin. Cubes with knotted holes, Trans. Amer. Math. Soc., 151:217–231, 1971. | DOI | MR | Zbl

[2] Michel Boileau. Uniformisation en dimension trois, Sém. Bourbaki, Exp. 855, Astérisque 266, 137–174, 2000. | Numdam | Zbl

[3] Francis Bonahon. Geometric Structures on 3-manifolds, in: Handbook of Geometric Topology (R.B. Daverman, R. Sher, Editors), 93–164, Elsevier, 2002. | DOI | Zbl

[4] Francis Bonahon and Laurence C. Siebenmann. New geometric splittings of classical knots and the classification and symmetries of arborescent knots, first version (around 1979) unpublished, revised version (June 12, 2010) http://www-bcf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf

[5] Steven Boyer. Dehn surgery on knots, in: Handbook of Geometric Topology (R.B. Daverman, R. Sher, Editors), 165–218, Elsevier, 2002. | DOI | Zbl

[6] Ryan Budney. JSJ decompositions of knot and link complements in S 3 , L’Enseignement Math., 52:319–359, 2006. | Zbl

[7] Gerhard Burde and Kunio Murasugi. Links and Seifert fiber spaces, Duke J. Math., 37:89–93, 1970. | DOI | MR | Zbl

[8] James W. Cannon and C.D. Feustel. Essential embeddings of annuli and Möbius bands in 3-manifolds, Trans. Amer. Math. Soc., 215:219–239, 1976. | DOI | Zbl

[9] Albrecht Dold. Lectures on algebraic topology, Springer, 1972. | DOI | Zbl

[10] David B.A. Epstein. Periodic flows on three-manifolds, Ann. Math., 95:66–82, 1972. | DOI | MR | Zbl

[11] C.D. Feustel. Some applications of Waldhausen’s results on irreducible surfaces, Trans. Amer. Math. Soc., 149:575–583, 1970. | DOI | MR | Zbl

[12] André Gramain. Rapport sur la théorie classique des noeuds (2ème partie), Sém. Bourbaki, exp. 732, Astérisque 201–203, 89–113, 1991. | Numdam | Zbl

[13] André Haefliger. Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, 16:367–397, 1962. | DOI | Zbl

[14] Allen Hatcher. Notes on basic 3-manifold topology, Course Notes, September 2000, http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html

[15] Pierre de la Harpe and Claude Weber, with an appendix by Denis Osin. Malnormal subgroups and Frobenius groups: basics and examples, Confl. Math., 6:65–76, 2014. | DOI | MR | Zbl

[16] John Hempel. 3–manifolds, Ann. Math. Studies, Princeton University Press, 1976. | Zbl

[17] William Jaco. Lectures on three-manifold topology, Regional Conference Series in Mathematics 43, Amer. Math. Soc., 1980. | DOI | Zbl

[18] William H. Jaco and Peter B. Shalen. Seifert fibered spaces in 3-manifolds, in: Geometric Topology (Proc. Georgia Top. Conf., Athens, Ga., 1977), 91–99, Academic Press, 1979. | DOI | Zbl

[19] William H. Jaco and Peter B. Shalen. Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21, 220(1), 1979.

[20] Klaus Johannson. Homotopy equivalences of 3-manifolds with boundary, Lecture Notes in Mathematics 761, Springer 1979. | DOI | Zbl

[21] Rinat Kashaev. On ring-valued invariants of topological pairs, preprint, 21 January 2007, arXiv:math/07015432v2 | Zbl

[22] Rinat Kashaev. Δ-groupoids in knot theory, Geom. Dedicata, 150:105–130, 2011. | DOI | MR | Zbl

[23] Jean-Louis Koszul. Sur certains groupes de transformations de Lie, in: Géométrie différentielle, Strasbourg, 26 mai – 1er juin 1953, 137–141, CNRS, 1953. | Zbl

[24] Walter D. Neumann and Gadde A. Swarup. Canonical decompositions of 3-manifolds, Geom. & Top., 1:21–40, 1997. | DOI | MR | Zbl

[25] Peter Orlik and Frank Raymond. Actions of SO(2) on 3-manifolds, in: Prof. Conf. Transform. Groups, New Orleans 1967, 297–318, Springer, 1968. | DOI

[26] Frank Raymond. Classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc., 131:51–78, 1968. | DOI | MR | Zbl

[27] Dale Rolfsen. Knots and links, Publish or Perish, 1976. | DOI | Zbl

[28] Horst Schubert. Knoten und Vollringe, Acta Math., 90(1):131–286, 1953. | DOI | MR | Zbl

[29] Peter Scott. The geometries of 3-manifolds, Bull. Lond. Math. Soc., 15:401–487, 1983, with errata on http://www.math.lsa.umich.edu/~pscott/ | DOI | MR | Zbl

[30] Herbert Seifert. Topologie dreidimensionaler gefaserter Räume, Acta Math., 60:147–288, 1933. Translated by W. Heil, appendix to [31], 359–422. | DOI | Zbl

[31] Herbert Seifert and William Threlfall. A textbook of topology, Academic Press, 1980. German original: Lehrbuch der Topologie, Teubner, 1934. | DOI

[32] Jonathan Simon. Roots and centralizers of peripheral elements in knot groups, Math. Ann., 222:205–209, 1976. | DOI | MR | Zbl

[33] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6:357–381, 1982. | DOI | MR | Zbl

[34] Friedhelm Waldhausen. Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I & II, Inv. Math., 3:308–333 & 4:87–117, 1967. | DOI | MR

[35] Friedhelm Waldhausen. On irreducible 3-manifolds which are sufficiently large, Ann. Math., 87:56–88, 1968. | DOI | MR | Zbl

[36] Friedhelm Waldhausen. On the determination of some bounded 3-manifolds by their fundamental groups alone, Proc. Int. Symposium on Top. and its Appl. (Herceg-Novi, Yugoslavia, 1968), 331–332, Beograd, 1969. | Zbl

[37] Friedhelm Waldhausen. Recent results on sufficiently large 3-manifolds, Proc. Symposia in Pure Math., 32:21–38, 1978. | DOI | Zbl

[38] Wilbur Whitten. Algebraic and geometric characterizations of knots, Inv. Math., 26:259–270, 1974. | DOI | MR | Zbl

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