The braided Thompson's groups are of type F

Kai-Uwe Bux 1 , Martin G. Fluch 2 , Marco Marschler 3 , Stefan Witzel 4 , and Matthew C. B. Zaremsky 5
  • 1 Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
  • 2 Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
  • 3 Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
  • 4 Mathematical Institute, University of Münster, 48149 Münster, Germany
  • 5 Department of Mathematical Sciences, Binghamton University,Binghamton, NY 13902, United States of America
Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel and Matthew C. B. Zaremsky

Abstract

We prove that the braided Thompson’s groups Vbr and Fbr are of type F, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected.

In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.

  • [1]

    Abels H. and Holz S., Higher generation by subgroups, J. Algebra 160 (1993), no. 2, 310–341.

    • Crossref
    • Export Citation
  • [2]

    Abramenko P. and Brown K. S., Buildings, Grad. Texts in Math. 248, Springer-Verlag, New York 2008.

  • [3]

    Athanasiadis C. A., Decompositions and connectivity of matching and chessboard complexes, Discrete Comput. Geom. 31 (2004), no. 3, 395–403.

    • Crossref
    • Export Citation
  • [4]

    Belk J. M. and Matucci F., Conjugacy and dynamics in Thompson’s groups, Geom. Dedicata 169 (2014), no. 1, 239–261.

    • Crossref
    • Export Citation
  • [5]

    Bestvina M. and Brady N., Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470.

    • Crossref
    • Export Citation
  • [6]

    Birman J. S., Braids, links, and mapping class groups, Ann. of Math. Stud. 82, Princeton University Press, Princeton 1974.

  • [7]

    Björner A., Lovász L., Vrećica S. T. and Živaljević R. T., Chessboard complexes and matching complexes, J. Lond. Math. Soc. (2) 49 (1994), no. 1, 25–39.

    • Crossref
    • Export Citation
  • [8]

    Brady T., Burillo J., Cleary S. and Stein M., Pure braid subgroups of braided Thompson’s groups, Publ. Mat. 52 (2008), no. 1, 57–89.

    • Crossref
    • Export Citation
  • [9]

    Bridson M. R. and Haefliger A., Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999.

  • [10]

    Brin M. G., The algebra of strand splitting. II. A presentation for the braid group on one strand, Internat. J. Algebra Comput. 16 (2006), no. 1, 203–219.

    • Crossref
    • Export Citation
  • [11]

    Brin M. G., The algebra of strand splitting. I. A braided version of Thompson’s group V, J. Group Theory 10 (2007), no. 6, 757–788.

  • [12]

    Brown K. S., Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), 45–75.

    • Crossref
    • Export Citation
  • [13]

    Brown K. S., The geometry of finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley 1989), Math. Sci. Res. Inst. Publ. 23, Springer-Verlag, New York (1992), 121–136.

  • [14]

    Brown K. S., The homology of Richard Thompson’s group F, Topological and asymptotic aspects of group theory, Contemp. Math. 394, American Mathematical Society, Providence (2006), 47–59.

  • [15]

    Burillo J. and Cleary S., Metric properties of braided Thompson’s groups, Indiana Univ. Math. J. 58 (2009), no. 2, 605–615.

    • Crossref
    • Export Citation
  • [16]

    Cannon J. W., Floyd W. J. and Parry W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3–4, 215–256.

  • [17]

    Charney R., The Deligne complex for the four-strand braid group, Trans. Amer. Math. Soc. 356 (2004), no. 10, 3881–3897.

  • [18]

    Degenhardt F., Endlichkeitseigenschaften gewisser Gruppen von Zöpfen unendlicher Ordnung, Ph.D. thesis, Johann Wolfgang Goethe-Universität, Frankfurt am Main 2000.

  • [19]

    Dehornoy P., The group of parenthesized braids, Adv. Math. 205 (2006), no. 2, 354–409.

    • Crossref
    • Export Citation
  • [20]

    Farb B. and Margalit D., A primer on mapping class groups, Princeton Math. Ser. 49, Princeton University Press, Princeton 2012.

  • [21]

    Farley D. S., Finiteness and CAT ( 0 ) $\rm CAT(0)$ properties of diagram groups, Topology 42 (2003), no. 5, 1065–1082.

    • Crossref
    • Export Citation
  • [22]

    Fluch M., Marschler M., Witzel S. and Zaremsky M. C. B., The Brin–Thompson groups sV are of type F $\text{F}_{\infty}$, Pacific J. Math. 266 (2013), no. 2, 283–295.

    • Crossref
    • Export Citation
  • [23]

    Funar L., Braided Houghton groups as mapping class groups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 53 (2007), no. 2, 229–240.

  • [24]

    Funar L. and Kapoudjian C., The braided Ptolemy–Thompson group is finitely presented, Geom. Topol. 12 (2008), no. 1, 475–530.

    • Crossref
    • Export Citation
  • [25]

    Funar L. and Kapoudjian C., The braided Ptolemy–Thompson group is asynchronously combable, Comment. Math. Helv. 86 (2011), no. 3, 707–768.

  • [26]

    Hatcher A., On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189–194.

    • Crossref
    • Export Citation
  • [27]

    Kassel C. and Turaev V., Braid groups, Grad. Texts in Math. 247, Springer-Verlag, New York 2008.

  • [28]

    Margalit D. and McCammond J., Geometric presentations for the pure braid group, J. Knot Theory Ramifications 18 (2009), no. 1, 1–20.

    • Crossref
    • Export Citation
  • [29]

    Meier J., Meinert H. and VanWyk L., Higher generation subgroup sets and the Σ-invariants of graph groups, Comment. Math. Helv. 73 (1998), no. 1, 22–44.

    • Crossref
    • Export Citation
  • [30]

    Putman A., Representation stability, congruence subgroups, and mapping class groups, preprint 2013, http://arxiv.org/abs/1201.4876v2.

  • [31]

    Quillen D., Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), no. 2, 101–128.

    • Crossref
    • Export Citation
  • [32]

    Spanier E. H., Algebraic topology, McGraw–Hill, New York 1966.

  • [33]

    Squier C. C., The homological algebra of Artin groups, Math. Scand. 75 (1994), no. 1, 5–43.

    • Crossref
    • Export Citation
  • [34]

    Stein M., Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), no. 2, 477–514.

    • Crossref
    • Export Citation
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