×

zbMATH — the first resource for mathematics

The braided Thompson’s groups are of type \(F_\infty\). (English) Zbl 1397.20053
Summary: We prove that the braided Thompson’s groups \(V_{\mathrm{br}}\) and \(F_{\mathrm{br}}\) are of type \(\mathrm{F}_{\infty}\), 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 H. Abels and S. Holz [J. Algebra 160, No. 2, 310–341 (1993; Zbl 0811.20036)].

MSC:
20F65 Geometric group theory
20F36 Braid groups; Artin groups
20J05 Homological methods in group theory
57M07 Topological methods in group theory
20E32 Simple groups
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abels H. and Holz S., Higher generation by subgroups, J. Algebra 160 (1993), no. 2, 310-341. · Zbl 0811.20036
[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. · Zbl 1061.05104
[4] Belk J. M. and Matucci F., Conjugacy and dynamics in Thompson’s groups, Geom. Dedicata 169 (2014), no. 1, 239-261. · Zbl 1321.20038
[5] Bestvina M. and Brady N., Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445-470. · Zbl 0888.20021
[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. · Zbl 0790.57014
[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. · Zbl 1185.20043
[9] Bridson M. R. and Haefliger A., Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999. · Zbl 0988.53001
[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. · Zbl 1170.20306
[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. · Zbl 1169.20021
[12] Brown K. S., Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), 45-75. · Zbl 0613.20033
[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. · Zbl 0753.20007
[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. · Zbl 1113.20043
[15] Burillo J. and Cleary S., Metric properties of braided Thompson’s groups, Indiana Univ. Math. J. 58 (2009), no. 2, 605-615. · Zbl 1200.20029
[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. · Zbl 0880.20027
[17] Charney R., The Deligne complex for the four-strand braid group, Trans. Amer. Math. Soc. 356 (2004), no. 10, 3881-3897. · Zbl 1077.20054
[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. · Zbl 1160.20027
[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 \(\rm CAT(0)\) properties of diagram groups, Topology 42 (2003), no. 5, 1065-1082. · Zbl 1044.20023
[22] Fluch M., Marschler M., Witzel S. and Zaremsky M. C. B., The Brin-Thompson groups sV are of type \(\text{F}_{∞}\), Pacific J. Math. 266 (2013), no. 2, 283-295. · Zbl 1292.20045
[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. · Zbl 1199.20069
[24] Funar L. and Kapoudjian C., The braided Ptolemy-Thompson group is finitely presented, Geom. Topol. 12 (2008), no. 1, 475-530. · Zbl 1187.20029
[25] Funar L. and Kapoudjian C., The braided Ptolemy-Thompson group is asynchronously combable, Comment. Math. Helv. 86 (2011), no. 3, 707-768. · Zbl 1266.57002
[26] Hatcher A., On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189-194. · Zbl 0727.57012
[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. · Zbl 1187.20048
[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. · Zbl 0899.57001
[30] Putman A., Representation stability, congruence subgroups, and mapping class groups, preprint 2013, .
[31] Quillen D., Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), no. 2, 101-128. · Zbl 0388.55007
[32] Spanier E. H., Algebraic topology, McGraw-Hill, New York 1966. · Zbl 0145.43303
[33] Squier C. C., The homological algebra of Artin groups, Math. Scand. 75 (1994), no. 1, 5-43. · Zbl 0839.20065
[34] Stein M., Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), no. 2, 477-514. · Zbl 0798.20025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.