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

Pierre de la Harpe[1]; Claude Weber[1]

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

Confluentes Mathematici (2014)

  • Volume: 6, Issue: 1, page 41-64
  • ISSN: 1793-7434

Abstract

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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 g P K g - 1 ∩ P K = { e } for any g ∈ G K with g ∉ 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.

How to cite

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de la Harpe, Pierre, and Weber, Claude. "On malnormal peripheral subgroups of the fundamental group of a $3$-manifold." Confluentes Mathematici 6.1 (2014): 41-64. <http://eudml.org/doc/275465>.

@article{delaHarpe2014,
abstract = {Let $K$ be a non-trivial knot in the $3$-sphere, $E_K$ its exterior, $G_K = \pi _1(E_K)$ its group, and $P_K = \pi _1(\partial E_K) \subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^\{-1\} \cap P_K = \lbrace e\rbrace $ 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.},
affiliation = {Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse; Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse},
author = {de la Harpe, Pierre, Weber, Claude},
journal = {Confluentes Mathematici},
keywords = {knot; knot group; peripheral subgroup; torus knot; cable knot; composite knot; malnormal subgroup; $3$-manifold; 3-manifold},
language = {eng},
number = {1},
pages = {41-64},
publisher = {Institut Camille Jordan},
title = {On malnormal peripheral subgroups of the fundamental group of a $3$-manifold},
url = {http://eudml.org/doc/275465},
volume = {6},
year = {2014},
}

TY - JOUR
AU - de la Harpe, Pierre
AU - Weber, Claude
TI - On malnormal peripheral subgroups of the fundamental group of a $3$-manifold
JO - Confluentes Mathematici
PY - 2014
PB - Institut Camille Jordan
VL - 6
IS - 1
SP - 41
EP - 64
AB - Let $K$ be a non-trivial knot in the $3$-sphere, $E_K$ its exterior, $G_K = \pi _1(E_K)$ its group, and $P_K = \pi _1(\partial E_K) \subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \cap P_K = \lbrace e\rbrace $ 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.
LA - eng
KW - knot; knot group; peripheral subgroup; torus knot; cable knot; composite knot; malnormal subgroup; $3$-manifold; 3-manifold
UR - http://eudml.org/doc/275465
ER -

References

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  1. R.H. Bing and Joseph M. Martin. Cubes with knotted holes, Trans. Amer. Math. Soc., 151:217–231, 1971. Zbl0213.25005MR278287
  2. Michel Boileau. Uniformisation en dimension trois, Sém. Bourbaki, Exp. 855, Astérisque 266, 137–174, 2000. Zbl0942.57013MR1772673
  3. Francis Bonahon. Geometric Structures on 3 -manifolds, in: Handbook of Geometric Topology (R.B. Daverman, R. Sher, Editors), 93–164, Elsevier, 2002. Zbl0997.57032MR1886669
  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. Zbl1058.57004MR1886670
  6. Ryan Budney. JSJ decompositions of knot and link complements in S 3 , L’Enseignement Math., 52:319–359, 2006. Zbl1114.57004MR2300613
  7. Gerhard Burde and Kunio Murasugi. Links and Seifert fiber spaces, Duke J. Math., 37:89–93, 1970. Zbl0195.54003MR253313
  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. Zbl0314.55004MR391094
  9. Albrecht Dold. Lectures on algebraic topology, Springer, 1972. Zbl0872.55001MR415602
  10. David B.A. Epstein. Periodic flows on three-manifolds, Ann. Math., 95:66–82, 1972. Zbl0231.58009MR288785
  11. C.D. Feustel. Some applications of Waldhausen’s results on irreducible surfaces, Trans. Amer. Math. Soc., 149:575–583, 1970. Zbl0203.25901MR261575
  12. André Gramain. Rapport sur la théorie classique des noeuds (2ème partie), Sém. Bourbaki, exp. 732, Astérisque 201–203, 89–113, 1991. Zbl0752.57003MR1157839
  13. André Haefliger. Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, 16:367–397, 1962. Zbl0122.40702MR189060
  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. Zbl1327.20030
  16. John Hempel. 3 –manifolds, Ann. Math. Studies, Princeton University Press, 1976. Zbl0345.57001MR415619
  17. William Jaco. Lectures on three-manifold topology, Regional Conference Series in Mathematics 43, Amer. Math. Soc., 1980. Zbl0433.57001MR565450
  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. Zbl0471.57001MR537728
  19. William H. Jaco and Peter B. Shalen. Seifert fibered spaces in 3 -manifolds, Mem. Amer. Math. Soc. 21, 220(1), 1979. Zbl0415.57005MR539411
  20. Klaus Johannson. Homotopy equivalences of 3 -manifolds with boundary, Lecture Notes in Mathematics 761, Springer 1979. Zbl0412.57007MR551744
  21. Rinat Kashaev. On ring-valued invariants of topological pairs, preprint, 21 January 2007, arXiv:math/07015432v2 Zbl1133.58003
  22. Rinat Kashaev. Δ -groupoids in knot theory, Geom. Dedicata, 150:105–130, 2011. Zbl1245.57015MR2753700
  23. Jean-Louis Koszul. Sur certains groupes de transformations de Lie, in: Géométrie différentielle, Strasbourg, 26 mai – 1er juin 1953, 137–141, CNRS, 1953. Zbl0101.16201
  24. Walter D. Neumann and Gadde A. Swarup. Canonical decompositions of 3 -manifolds, Geom. & Top., 1:21–40, 1997. Zbl0886.57009MR1469066
  25. Peter Orlik and Frank Raymond. Actions of S O ( 2 ) on 3 -manifolds, in: Prof. Conf. Transform. Groups, New Orleans 1967, 297–318, Springer, 1968. Zbl0172.25402MR263112
  26. Frank Raymond. Classification of the actions of the circle on 3 -manifolds, Trans. Amer. Math. Soc., 131:51–78, 1968. Zbl0157.30602MR219086
  27. Dale Rolfsen. Knots and links, Publish or Perish, 1976. Zbl0854.57002MR515288
  28. Horst Schubert. Knoten und Vollringe, Acta Math., 90(1):131–286, 1953. Zbl0051.40403MR72482
  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/ Zbl0561.57001MR705527
  30. Herbert Seifert. Topologie dreidimensionaler gefaserter Räume, Acta Math., 60:147–288, 1933. Translated by W. Heil, appendix to [31], 359–422. Zbl0006.08304MR1555366
  31. Herbert Seifert and William Threlfall. A textbook of topology, Academic Press, 1980. German original: Lehrbuch der Topologie, Teubner, 1934. MR575168
  32. Jonathan Simon. Roots and centralizers of peripheral elements in knot groups, Math. Ann., 222:205–209, 1976. Zbl0314.55003MR418079
  33. William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6:357–381, 1982. Zbl0496.57005MR648524
  34. Friedhelm Waldhausen. Eine Klasse von 3 -dimensionalen Mannigfaltigkeiten I & II, Inv. Math., 3:308–333 & 4:87–117, 1967. Zbl0168.44503MR235576
  35. Friedhelm Waldhausen. On irreducible 3 -manifolds which are sufficiently large, Ann. Math., 87:56–88, 1968. Zbl0157.30603MR224099
  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. Zbl0202.54702
  37. Friedhelm Waldhausen. Recent results on sufficiently large 3 -manifolds, Proc. Symposia in Pure Math., 32:21–38, 1978. Zbl0391.57011MR520520
  38. Wilbur Whitten. Algebraic and geometric characterizations of knots, Inv. Math., 26:259–270, 1974. Zbl0291.55004MR365548

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