Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-28T04:21:07.436Z Has data issue: false hasContentIssue false
  • English
  • Français

VIRTUALLY FIBERING RIGHT-ANGLED COXETER GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  23 August 2019

Kasia Jankiewicz ,
Sergey Norin  and
Daniel T. Wise
Kasia Jankiewicz
Affiliation:
Department of Mathematics and Statsistics, McGill University, Montreal, Quebec, CanadaH3A 0B9 (kasia@math.uchicago.edu; snorine@gmail.com; wise@math.mcgill.ca)
Sergey Norin
Affiliation:
Department of Mathematics and Statsistics, McGill University, Montreal, Quebec, CanadaH3A 0B9 (kasia@math.uchicago.edu; snorine@gmail.com; wise@math.mcgill.ca)
Daniel T. Wise
Affiliation:
Department of Mathematics and Statsistics, McGill University, Montreal, Quebec, CanadaH3A 0B9 (kasia@math.uchicago.edu; snorine@gmail.com; wise@math.mcgill.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the 120-cell or the 24-cell.

Keywords

Coxeter groupsMorse theoryCoherent groups

MSC classification

Primary: 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 957 - 987
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by NSERC.

References

Agol, I., Criteria for virtual fibering, J. Topol. 1(2) (2008), 269–284.CrossRefGoogle Scholar
Agol, I., The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning.Google Scholar
Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M. L., Shapiro, M. and Short, H., Notes on word hyperbolic groups, in Group Theory from a Geometrical Viewpoint (Trieste, 1990), (ed. Ghys, É., Haefliger, A. and Verjovsky, A.), pp. 3–63 (World Sci. Publishing, River Edge, NJ, 1991). Edited by H. Short.Google Scholar
Andreev, E. M., Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83(125) (1970), 256–260.Google Scholar
Bestvina, M. and Brady, N., Morse theory and finiteness properties of groups, Invent. Math. 129(3) (1997), 445–470.CrossRefGoogle Scholar
Bieri, R., Normal subgroups in duality groups and in groups of cohomological dimension 2, J. Pure Appl. Algebra 7(1) (1976), 35–51.CrossRefGoogle Scholar
Bieri, R., Homological dimension of discrete groups, 2nd ed. (Queen Mary College Department of Pure Mathematics, London, 1981).Google Scholar
Bollobás, B., The distribution of the maximum degree of a random graph, Discrete Math. 32(2) (1980), 201–203.CrossRefGoogle Scholar
Bowditch, B. H. and Mess, G., A 4-dimensional Kleinian group, Trans. Amer. Math. Soc. 344(1) (1994), 391–405.Google Scholar
Brady, N., Branched coverings of cubical complexes and subgroups of hyperbolic groups, J. Lond. Math. Soc. (2) 60(2) (1999), 461–480.CrossRefGoogle Scholar
Canary, R. D., A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35(3) (1996), 751–778.CrossRefGoogle Scholar
Caprace, P.-E., Buildings with isolated subspaces and relatively hyperbolic Coxeter groups, Innov. Incidence Geom. 10 (2009), 15–31.CrossRefGoogle Scholar
Caprace, P.-E., Buildings with isolated subspaces and relatively hyperbolic Coxeter groups, Preprint, 2013, arXiv:0703799v3, pp. 1–12.Google Scholar
Davis, M. W., A hyperbolic 4-manifold, Proc. Amer. Math. Soc. 93(2) (1985), 325–328.Google Scholar
Davis, M. W., The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, Volume 32 (Princeton University Press, Princeton, NJ, 2008).Google Scholar
Davis, M. W. and Okun, B., Vanishing theorems and conjectures for the ℓ 2 -homology of right-angled Coxeter groups, Geom. Topol. 5 (2001), 7–74.CrossRefGoogle Scholar
Erdoős, P. and Rényi, A., On the existence of a factor of degree one of a connected random graph, Acta Math. Acad. Sci. Hungar. 17 (1966), 359–368.CrossRefGoogle Scholar
Gould, R. J., Recent advances on the Hamiltonian problem: Survey III, Graphs Combin. 30(1) (2014), 1–46.CrossRefGoogle Scholar
Grinberg, È. J., Plane homogeneous graphs of degree three without Hamiltonian circuits, in Latvian Math. Yearbook, 4 (Russian), pp. 51–58 (Izdat. ‘Zinatne’, Riga, 1968).Google Scholar
Hodgson, C. D., Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, in Topology ’90 (Columbus, OH, 1990), Ohio State Univ. Math. Res. Inst. Publ., Volume 1, pp. 185–193 (de Gruyter, Berlin, 1992).Google Scholar
Inoue, T., Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8(3) (2008), 1523–1565.CrossRefGoogle Scholar
Jankiewicz, K. and Wise, D. T., Incoherent Coxeter groups, Proc. Amer. Math. Soc. 144(5) (2016), 1857–1866.CrossRefGoogle Scholar
Kapovich, M., Noncoherence of arithmetic hyperbolic lattices, Geom. Topol. 17(1) (2013), 39–71.CrossRefGoogle Scholar
Kapovich, M., Potyagailo, L. and Vinberg, E., Noncoherence of some lattices in Isom(ℍn), in The Zieschang Gedenkschrift, Geom. Topol. Monogr., Volume 14, pp. 335–351 (Geom. Topol. Publ., Coventry, 2008).Google Scholar
Liu, Y., Virtual cubulation of nonpositively curved graph manifolds, J. Topol. 6(4) (2013), 793–822.CrossRefGoogle Scholar
Lutz, F. H. and Nevo, E., Stellar theory for flag complexes, Preprint, 2014, arXiv:1302.5197v3, pp. 1–12.Google Scholar
Palásti, I., On the connectedness of bichromatic random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1964), 431–441.Google Scholar
Pogorelov, A. V., Regular decomposition of the Lobačevskiĭ space, Mat. Zametki 1 (1967), 3–8.Google Scholar
Pontiveros, G. F., Glebov, R. and Karpas, I., Virtually fibering random right-angled Coxeter groups, pp. 1–11, 2016.Google Scholar
Przytycki, P. and Wise, D. T., Mixed 3-manifolds are virtually special, pp. 1–24. Available at arXiv:1205.6742.Google Scholar
Rivin, I., On geometry of convex polyhedra in hyperbolic 3-space, PhD Thesis, Princeton University, ProQuest LLC, Ann Arbor, MI, 1986.Google Scholar
Roeder, R. K. W., Hubbard, J. H. and Dunbar, W. D., Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57(3) (2007), 825–882.CrossRefGoogle Scholar
Stallings, J., On fibering certain 3-manifolds, in Topology of 3-manifolds and Related Topics (Proc. The Univ. of Georgia Institute, 1961), pp. 95–100 (Prentice-Hall, Englewood Cliffs, NJ, 1962).Google Scholar
Stallings, J., A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541–543.CrossRefGoogle Scholar
Thurston, W. P., The geometry and topology of $3$ -manifolds, Lecture Notes.Google Scholar
Vesnin, A. Yu., Three-dimensional hyperbolic manifolds of Löbell type, Sibirsk. Mat. Zh. 28(5) (1987), 50–53.Google Scholar
Wikipedia. 600-cell — Wikipedia, the free encyclopedia, 2017. [Online; accessed 12-April-2017].Google Scholar
Wise, D. T., The structure of groups with a quasiconvex hierarchy, Ann. of Math (2), to appear.Google Scholar
Wise, D. T., Morse theory, random subgraphs, and incoherent groups, Bull. Lond. Math. Soc. 43(5) (2011), 840–848.CrossRefGoogle Scholar