On the Tits alternative for $\protect \mathit{PD}(3)$ groups

Michel Boileau; Steven Boyer

doi:10.5802/afst.1604

On the Tits alternative for

PD (3)

groups

Michel Boileau ¹ ; Steven Boyer ²

¹ Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France
² Département de Mathématiques, Université du Québec à Montréal, 201 avenue du Président-Kennedy, Montréal, QC H2X 3Y7, Canada

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 3, pp. 397-415.

Résumé
Abstract

On montre qu’un groupe à dualité de Poincaré de dimension $3$ , presque cohérent et tel que tout sous-groupe d’indice fini contienne un sous-groupe propre de type fini non cyclique, vérifie l’alternative de Tits. On obtient en particulier qu’un groupe à dualité de Poincaré de dimension $3$ , presque cohérent et qui ne peut pas être engendré par moins de $4$ éléments, contient toujours un groupe libre non abélien.

We prove the Tits alternative for an almost coherent $PD (3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD (3)$ group which cannot be generated by less than four elements always contains a rank 2 free group.

Publié le : 2019-12-06

DOI : 10.5802/afst.1604

Affiliations des auteurs :

Michel Boileau ¹ ; Steven Boyer ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Michel Boileau and Steven Boyer},
     title = {On the {Tits} alternative for $\protect \mathit{PD}(3)$ groups},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {397--415},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {3},
     year = {2019},
     doi = {10.5802/afst.1604},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1604/}
}

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Michel Boileau; Steven Boyer. On the Tits alternative for $\protect \mathit{PD}(3)$ groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 3, pp. 397-415. doi : 10.5802/afst.1604. https://afst.centre-mersenne.org/articles/10.5802/afst.1604/

Bibliographie
Cité par

[1] Alejandro Adem; R. James Milgram Cohomology of Finite Groups, Grundlehren der Mathematischen Wissenschaften, 309, Springer, 2004 | MR | Zbl

[2] Ian Agol The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 | MR | Zbl

[3] Selman Akbulut; John McCarthy Casson’s invariant for oriented homology 3-spheres. An exposition, Mathematical Notes, 36, Princeton University Press, 1990 | Zbl

[4] Louis Auslander; F. E. A. Johnson On a Conjecture of C.T.C. Wall, J. Lond. Math. Soc., Volume 14 (1976), pp. 331-332 | DOI | MR | Zbl

[5] Gilbert Baumslag Topics in combinatorial group theory, Lectures in Mathematics, Birkhäuser, 1993 | Zbl

[6] Gilbert Baumslag; Peter B. Shalen Amalgamated products and finitely presented groups, Comment. Math. Helv., Volume 65 (1990) no. 2, pp. 243-254 | DOI | MR | Zbl

[7] Leila Ben Abdelghani; Steven Boyer A calculation of the Culler-Shalen seminorms associated to small Seifert Dehn fillings, Proc. Lond. Math. Soc., Volume 83 (2001) no. 1, pp. 235-256 | DOI | MR | Zbl

[8] Laurent Bessières; Gérard Besson; Sylvain Maillot; Michel Boileau; Joan Porti Geometrisation of $3$ -manifolds, EMS Tracts in Mathematics, 13, European Mathematical Society, 2010 | MR | Zbl

[9] Robert Bieri Homological dimension of discrete groups, Queen Mary College Mathematics Notes, Queen Mary College, 1976 | Zbl

[10] Robert Bieri; Beno Eckmann Finiteness Properties of Duality groups, Comment. Math. Helv., Volume 49 (1974), pp. 74-83 | DOI | MR | Zbl

[11] Robert Bieri; Beno Eckmann Relative homology and Poincaré duality for groups pairs, J. Pure Appl. Algebra, Volume 13 (1978), pp. 277-319 | DOI | Zbl

[12] Robert Bieri; Ralph Strebel Almost finitely presented soluble groups, Comment. Math. Helv., Volume 53 (1978), pp. 258-278 | DOI | MR | Zbl

[13] Brian Bowditch Planar groups and the Seifert conjecture, J. Reine Angew. Math., Volume 576 (2004), pp. 11-62 | MR | Zbl

[14] Brian Bowditch; Brian Bowditch A 4-dimensional Kleinian group, Trans. Am. Math. Soc., Volume 344 (1994), pp. 391-405 | MR | Zbl

[15] Steven Boyer; Marc Culler; Peter B. Shalen; Xingru Zhang Characteristic subsurfaces, character varieties and Dehn fillings, Geom. Topol., Volume 12 (2008) no. 1, pp. 233-297 | DOI | MR | Zbl

[16] Steven Boyer; Xingru Zhang Finite Dehn surgery on knots, J. Am. Math. Soc., Volume 9 (1996), pp. 1005-1050 | DOI | MR | Zbl

[17] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1982 | Zbl

[18] Henri Cartan; Samuel Eilenberg Homological Algebra, Princeton Mathematical Series, 19, Princeton University Press, 1956 | Zbl

[19] Fabrice Castel Centralisateurs d’éléments dans les $P D (3)$ paires, Comment. Math. Helv., Volume 82 (2007) no. 3, pp. 499-517 | DOI | MR | Zbl

[20] Michael W. Davis The cohomology of a Coxeter group with group ring coefficients, Duke Math. J., Volume 91 (1998) no. 2, pp. 297-314 | DOI | MR | Zbl

[21] Martin J. Dunwoody; Eric L. Swenson The algebraic torus theorem, Invent. Math., Volume 140 (2000) no. 3, pp. 605-637 | DOI | MR | Zbl

[22] Beno Eckmann Poincaré duality groups of dimension two are surface groups, Combinatorial group theory and topology (Annals of Mathematics Studies), Volume 111, Princeton University Press, 1987, pp. 35-51 | DOI | MR | Zbl

[23] Benny Evans; Louise Moser Solvable fundamental groups of compact 3-manifolds, Trans. Am. Math. Soc., Volume 168 (1972), pp. 189-210 | MR | Zbl

[24] Daciberg L. Gonçalves; Mauro Spreafico; Oziride Manzoli Neto The Borsuk-Ulam theorem for homotopy spherical space forms, J. Fixed Point Theory Appl., Volume 9 (2011) no. 2, pp. 285-294 | DOI | MR | Zbl

[25] Francisco González Acuña Homomorphs of knot groups, Ann. Math., Volume 102 (1975), pp. 373-377 | DOI | MR | Zbl

[26] Daniel Groves; Jason Fox Manning; Henry Wilton Recognizing geometric $3$ -manifold groups using the word problem (2012) (https://arxiv.org/abs/1210.2101)

[27] Jonathan A. Hillman Seifert fibre spaces and Poincaré duality groups, Math. Z., Volume 190 (1985), pp. 365-369 | DOI | MR | Zbl

[28] Jonathan A. Hillman Three dimensional Poincaré duality groups which are extensions, Math. Z., Volume 195 (1987), pp. 89-92 | DOI | Zbl

[29] Jonathan A. Hillman Four-manifolds, geometries and knots, Geometry and Topology Monographs, 5, Geometry and Topology Publications, 2002 | MR | Zbl

[30] Jonathan A. Hillman Tits alternatives and low dimensional topology, J. Math. Soc. Japan, Volume 55 (2003) no. 2, pp. 365-383 | DOI | MR | Zbl

[31] Michael Kapovich; Bruce Kleiner Coarse Alexander duality and duality groups, J. Differ. Geom., Volume 69 (2005) no. 2, pp. 279-352 | DOI | MR | Zbl

[32] Bruce Kleiner; John Lott Notes on Perelman’s papers, Geom. Topol., Volume 12 (2008) no. 5, pp. 2587-2858 | DOI | MR | Zbl

[33] Dessislava H. Kochloukova; Pavel A. Zalesskii Tits alternative for 3-manifold groups, Arch. Math., Volume 88 (2007) no. 4, pp. 364-367 | DOI | MR | Zbl

[34] László G. Kovács On finite soluble groups, Math. Z., Volume 103 (1968), pp. 37-39 | DOI | MR | Zbl

[35] Aleksandr G. Kurosh The theory of groups. Vol. 1, Chelsea Publishing, 1960

[36] Michel Lazard Groupes analytiques p-adiques, Publ. Math., Inst. Hautes Étud. Sci., Volume 26 (1965), pp. 389-603 | MR | Zbl

[37] Yi Liu Virtual cubulation of nonpositively curved graph manifolds, J. Topol., Volume 6 (2013) no. 4, pp. 793-822 | DOI | MR | Zbl

[38] Alexander Lubotzky A group-theoretic characterisation of linear groups, J. Algebra, Volume 113 (1988) no. 1, pp. 207-214 | DOI | MR | Zbl

[39] Alexander Lubotzky; Avinoam Mann Residually finite groups of finite rank, Proc. Camb. Philos. Soc., Volume 106 (1989) no. 3, pp. 385-388 | DOI | MR | Zbl

[40] Roger Lyndon Two notes on Rankin’s book on the modular group, J. Aust. Math. Soc., Volume 16 (1973), pp. 454-457 | DOI | MR | Zbl

[41] Roger Lyndon; Paul Schupp Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, Springer, 1977 | MR | Zbl

[42] Geoffrey Mess Examples of Poincaré duality groups, Proc. Am. Math. Soc., Volume 110 (1990) no. 4, pp. 1145-1146 | MR | Zbl

[43] Geoffrey Mess Finite covers of 3-manifolds, and a theorem of Lubotzky (1990) (preprint) | Zbl

[44] Alexander Yu. Olʼshanskii An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 43 (1979), pp. 1328-1393 translated in Math. USSR-Izv. 15 (1980), 531-588 | MR | Zbl

[45] Walter Parry A sharper Tits alternative for 3-manifold groups, Isr. J. Math., Volume 77 (1992) no. 3, p. 365-271 | MR | Zbl

[46] Leonid Potyagailo Finitely generated Kleinian groups in 3-space and $3$ -manifolds of infinite homotopy type, Trans. Am. Math. Soc., Volume 344 (1994) no. 1, pp. 57-77 | MR | Zbl

[47] Piotr Przytycki; Daniel T. Wise Mixed $3$ -manifolds are virtually special, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 318-347 | MR | Zbl

[48] Alexander Reznikov Three-manifolds class field theory. (Homology of coverings for a nonvirtually $b_{1}$ -positive manifold), Sel. Math., New Ser., Volume 3 (1997) no. 3, pp. 361-399 | DOI | MR | Zbl

[49] Peter Scott Finitely generated $3$ -manifold groups are finitely presented, J. Lond. Math. Soc., Volume 6 (1973), pp. 437-440 | DOI | MR | Zbl

[50] Peter Scott The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983), pp. 401-487 | DOI | MR | Zbl

[51] Peter Scott; Gadde A. Swarup Regular neighbourhoods and canonical decompositions for groups, Astérisque, 289, Société Mathématique de France, 2003 | Zbl

[52] Dan Segal A footnote on residually finite groups, Isr. J. Math., Volume 94 (1996), pp. 1-5 | DOI | MR | Zbl

[53] Peter B. Shalen; Philip Wagreich Growth rates, $ℤ_{p}$ -homology, and volumes of hyperbolic $3$ -manifolds, Trans. Am. Math. Soc., Volume 331 (1992) no. 2, pp. 895-917 | Zbl

[54] John Stallings Homology and central series of groups, J. Algebra, Volume 2 (1965), pp. 170-181 | DOI | MR | Zbl

[55] John Stallings Group theory and three-dimensional manifolds, Yale Mathematical Monographs, 4, Yale University Press, 1971 | MR | Zbl

[56] Ralph Strebel A remark on subgroups of infinite index in Poincaré duality groups, Comment. Math. Helv., Volume 52 (1977), pp. 317-324 | DOI | Zbl

[57] Richard G. Swan A new method in fixed point theory, Comment. Math. Helv., Volume 34 (1960), pp. 1-16 | DOI | MR | Zbl

[58] Charles B. Thomas Splitting theorems for certain $P D (3)$ groups, Math. Z., Volume 186 (1984), pp. 201-209 | DOI | MR | Zbl

[59] Charles B. Thomas Elliptic structures on $3$ -manifolds, London Mathematical Society Lecture Note Series, 104, Cambridge University Press, 1986 | MR | Zbl

[60] Charles B. Thomas $3$ -manifolds and $P D (3)$ -groups, Novikov conjectures, index theorems and rigidity. Vol. 2 (London Mathematical Society Lecture Note Series), Volume 227, Cambridge University Press, 1995, pp. 301-308 | DOI | MR | Zbl

[61] Jacques Tits Free subgroups in linear groups, J. Algebra, Volume 20 (1972), pp. 250-270 | DOI | MR | Zbl

[62] Vladimir G. Turaev Fundamental groups of three-dimensional manifolds and Poincaré duality, Proc. Steklov Inst. Math., Volume 4 (1984), pp. 249-257 | Zbl

[63] Charles T. C. Wall The geometry of abstract groups and their splittings, Rev. Mat. Complut., Volume 16 (2003) no. 1, pp. 5-101 | MR | Zbl

[64] Charles T. C. Wall Poincaré duality in dimension 3, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7, Geometry and Topology Publications, 2004, pp. 1-26 | Zbl

[65] Daniel T. Wise The structure of groups with a quasiconvex hierarchy http://www.math.mcgill.ca/wise/papers.html (accessed 2012-10-29), to appear in Annals of Mathematics Studies | Zbl

[66] Daniel T. Wise The structure of groups with a quasi-convex hierarchy, Electron. Res. Announc. Math. Sci., Volume 16 (2009), pp. 44-55 | Zbl

[67] Joseph Wolf Spaces of Constant Curvature, Publish or Perish, 1984 | Zbl

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