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Mixed $ 3$-manifolds are virtually special


Authors: Piotr Przytycki and Daniel T. Wise
Journal: J. Amer. Math. Soc. 31 (2018), 319-347
MSC (2010): Primary 20F65, 57M50
DOI: https://doi.org/10.1090/jams/886
Published electronically: October 19, 2017
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Abstract: Let $ M$ be a compact oriented irreducible $ 3$-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that $ \pi _1M$ is virtually special.


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  • [Ago04] Ian Agol, Tameness of hyperbolic 3-manifolds (2004), available at arXix:GT/0405568.
  • [Ago08] Ian Agol, Criteria for virtual fibering, J. Topol. 1 (2008), no. 2, 269-284. MR 2399130, https://doi.org/10.1112/jtopol/jtn003
  • [Ago13] Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045-1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553
  • [AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton, 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015. MR 3444187
  • [BW13] Hadi Bigdely and Daniel T. Wise, Quasiconvexity and relatively hyperbolic groups that split, Michigan Math. J. 62 (2013), no. 2, 387-406. MR 3079269, https://doi.org/10.1307/mmj/1370870378
  • [Bon86] Francis Bonahon, Bouts des variétés hyperboliques de dimension $ 3$, Ann. of Math. (2) 124 (1986), no. 1, 71-158 (French). MR 847953, https://doi.org/10.2307/1971388
  • [Bon02] Francis Bonahon, Geometric structures on 3-manifolds, Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, pp. 93-164. MR 1886669
  • [Bri01] Martin R. Bridson, On the subgroups of semihyperbolic groups, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 85-111. MR 1929323
  • [CG06] Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385-446. MR 2188131, https://doi.org/10.1090/S0894-0347-05-00513-8
  • [Can96] Richard D. Canary, A covering theorem for hyperbolic $ 3$-manifolds and its applications, Topology 35 (1996), no. 3, 751-778. MR 1396777, https://doi.org/10.1016/0040-9383(94)00055-7
  • [CLR97] D. Cooper, D. D. Long, and A. W. Reid, Essential closed surfaces in bounded $ 3$-manifolds, J. Amer. Math. Soc. 10 (1997), no. 3, 553-563. MR 1431827, https://doi.org/10.1090/S0894-0347-97-00236-1
  • [DJ00] Michael W. Davis and Tadeusz Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000), no. 3, 229-235. MR 1783167, https://doi.org/10.1016/S0022-4049(99)00175-9
  • [GP16] Victor Gerasimov and Leonid Potyagailo, Quasiconvexity in relatively hyperbolic groups, J. Reine Angew. Math. 710 (2016), 95-135. MR 3437561, https://doi.org/10.1515/crelle-2015-0029
  • [HW08] Frédéric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551-1620. MR 2377497, https://doi.org/10.1007/s00039-007-0629-4
  • [HW10] Frédéric Haglund and Daniel T. Wise, Coxeter groups are virtually special, Adv. Math. 224 (2010), no. 5, 1890-1903. MR 2646113, https://doi.org/10.1016/j.aim.2010.01.011
  • [Ham01] Emily Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic $ n$-orbifolds, Proc. London Math. Soc. (3) 83 (2001), no. 3, 626-646. MR 1851085, https://doi.org/10.1112/plms/83.3.626
  • [Hru10] G. Christopher Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), no. 3, 1807-1856. MR 2684983, https://doi.org/10.2140/agt.2010.10.1807
  • [HW14] G. C. Hruska and Daniel T. Wise, Finiteness properties of cubulated groups, Compos. Math. 150 (2014), no. 3, 453-506. MR 3187627, https://doi.org/10.1112/S0010437X13007112
  • [HW99] Tim Hsu and Daniel T. Wise, On linear and residual properties of graph products, Michigan Math. J. 46 (1999), no. 2, 251-259. MR 1704150, https://doi.org/10.1307/mmj/1030132408
  • [Jan17] Kasia Jankiewicz, The fundamental theorem of cubical small cancellation theory, Trans. Amer. Math. Soc. 369 (2017), no. 6, 4311-4346. MR 3624411, https://doi.org/10.1090/tran/6852
  • [Kap01] Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613
  • [KM04] P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU$ (2)$, Math. Res. Lett. 11 (2004), no. 5-6, 741-754. MR 2106239, https://doi.org/10.4310/MRL.2004.v11.n6.a3
  • [Lee95] Bernhard Leeb, $ 3$-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995), no. 2, 277-289. MR 1358977, https://doi.org/10.1007/BF01231445
  • [Liu13] Yi Liu, Virtual cubulation of nonpositively curved graph manifolds, J. Topol. 6 (2013), no. 4, 793-822. MR 3145140, https://doi.org/10.1112/jtopol/jtt010
  • [MP09] Eduardo Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009), no. 2, 317-342. MR 2486802, https://doi.org/10.4171/GGD/59
  • [Min06] Ashot Minasyan, Separable subsets of GFERF negatively curved groups, J. Algebra 304 (2006), no. 2, 1090-1100. MR 2264291, https://doi.org/10.1016/j.jalgebra.2006.03.050
  • [PW14] Piotr Przytycki and Daniel T. Wise, Graph manifolds with boundary are virtually special, J. Topol. 7 (2014), no. 2, 419-435. MR 3217626, https://doi.org/10.1112/jtopol/jtt009
  • [Rol98] Martin Roller, Poc sets, median algebras, and group actions. An extended study of Dunwoody's construction and Sageev's theorem. (1998), available at www.personal.soton.ac.uk/gan/Roller.pdf.
  • [RW98] J. Hyam Rubinstein and Shicheng Wang, $ \pi_1$-injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998), no. 4, 499-515. MR 1639876, https://doi.org/10.1007/s000140050066
  • [Thu80] William P. Thurston, The Geometry and Topology of Three-Manifolds (1980), Princeton University course notes, available at http://www.msri.org/publications/books/gt3m/.
  • [Thu86] William P. Thurston, A norm for the homology of $ 3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i-vi and 99-130. MR 823443
  • [Wis12] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, vol. 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012. MR 2986461
  • [Wis17] Daniel T. Wise, The structure of groups with quasiconvex hierarchy, Ann. of Math. Stud. to appear.

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Additional Information

Piotr Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland; and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9
Email: piotr.przytycki@mcgill.ca

Daniel T. Wise
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9
Email: wise@math.mcgill.ca

DOI: https://doi.org/10.1090/jams/886
Received by editor(s): June 6, 2012
Received by editor(s) in revised form: July 24, 2013, May 14, 2014, and May 12, 2017
Published electronically: October 19, 2017
Additional Notes: The first author was partially supported by MNiSW grant N201 012 32/0718, the Foundation for Polish Science, National Science Centre DEC-2012/06/A/ST1/00259 and UMO-2015/18/M/ST1/00050, NSERC and FRQNT
The second author was supported by NSERC
Article copyright: © Copyright 2017 American Mathematical Society

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