The 4-string braid group B 4 has property RD and exponential mesoscopic rank

Sylvain Barré; Mikaël Pichot

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 4, page 479-502
  • ISSN: 0037-9484

Abstract

top
We prove that the braid group B 4 on 4 strings, its central quotient B 4 / z , and the automorphism group Aut ( F 2 ) of the free group F 2 on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group B 4 is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

How to cite

top

Barré, Sylvain, and Pichot, Mikaël. "The 4-string braid group $B_4$ has property RD and exponential mesoscopic rank." Bulletin de la Société Mathématique de France 139.4 (2011): 479-502. <http://eudml.org/doc/272640>.

@article{Barré2011,
abstract = {We prove that the braid group $B_4$ on 4 strings, its central quotient $B_4/\langle z\rangle $, and the automorphism group $\operatorname\{Aut\}(F_2)$ of the free group $F_2$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_4$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.},
author = {Barré, Sylvain, Pichot, Mikaël},
journal = {Bulletin de la Société Mathématique de France},
keywords = {braid groups; property RD; CAT(0) spaces},
language = {eng},
number = {4},
pages = {479-502},
publisher = {Société mathématique de France},
title = {The 4-string braid group $B_4$ has property RD and exponential mesoscopic rank},
url = {http://eudml.org/doc/272640},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Barré, Sylvain
AU - Pichot, Mikaël
TI - The 4-string braid group $B_4$ has property RD and exponential mesoscopic rank
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 4
SP - 479
EP - 502
AB - We prove that the braid group $B_4$ on 4 strings, its central quotient $B_4/\langle z\rangle $, and the automorphism group $\operatorname{Aut}(F_2)$ of the free group $F_2$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_4$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.
LA - eng
KW - braid groups; property RD; CAT(0) spaces
UR - http://eudml.org/doc/272640
ER -

References

top
  1. [1] W. Ballmann & M. Brin – « Rank rigidity of Euclidean polyhedra », Amer. J. Math.122 (2000), p. 873–885. Zbl1041.53029MR1781923
  2. [2] S. Barré & M. Pichot – « Sur les immeubles triangulaires et leurs automorphismes », Geom. Dedicata130 (2007), p. 71–91. Zbl1142.51011MR2365779
  3. [3] —, « Friezes in polyhedral complexes and application », in preparation. 
  4. [4] —, « Intermediate rank and property RD », preprint arXiv:0710.1514. 
  5. [5] —, « Property RD for D. Wise non Hopfian group », preprint. 
  6. [6] J. Behrstock & Y. Minsky – « Centroids and the rapid decay property in mapping class groups », preprint arXiv:0810.1969. Zbl1269.20032MR2855801
  7. [7] T. Brady – « Artin groups of finite type with three generators », Michigan Math. J.47 (2000), p. 313–324. Zbl0996.20022MR1793627
  8. [8] T. Brady & J. P. McCammond – « Three-generator Artin groups of large type are biautomatic », J. Pure Appl. Algebra151 (2000), p. 1–9. Zbl1004.20023MR1770639
  9. [9] M. R. Bridson & A. Haefliger – Metric spaces of non-positive curvature, Grundl. Math. Wiss., vol. 319, Springer, 1999. Zbl0988.53001MR1744486
  10. [10] E. Brieskorn & K. Saito – « Artin-Gruppen und Coxeter-Gruppen », Invent. Math.17 (1972), p. 245–271. Zbl0243.20037MR323910
  11. [11] G. Burde & H. Zieschang – Knots, second éd., de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., 2003. Zbl1009.57003MR1959408
  12. [12] R. Charney – « Artin groups of finite type are biautomatic », Math. Ann.292 (1992), p. 671–683. Zbl0736.57001MR1157320
  13. [13] —, « Geodesic automation and growth functions for Artin groups of finite type », Math. Ann.301 (1995), p. 307–324. Zbl0813.20042MR1314589
  14. [14] —, « The Deligne complex for the four-strand braid group », Trans. Amer. Math. Soc.356 (2004), p. 3881–3897. Zbl1077.20054MR2058510
  15. [15] I. Chatterji – « Property (RD) for cocompact lattices in a finite product of rank one Lie groups with some rank two Lie groups », Geom. Dedicata96 (2003), p. 161–177. Zbl1012.22018MR1956838
  16. [16] I. Chatterji, C. Pittet & L. Saloff-Coste – « Connected Lie groups and property RD », Duke Math. J.137 (2007), p. 511–536. Zbl1119.22006MR2309152
  17. [17] I. Chatterji & K. Ruane – « Some geometric groups with rapid decay », Geom. Funct. Anal.15 (2005), p. 311–339. Zbl1134.22005MR2153902
  18. [18] I. Chatterji & L. Saloff-Coste – « Property of rapid decay. List of open problems from the 2006 workshop held at the AIM, Palo Alto, California », http://www.aimath.org/WWN/rapiddecay, 2006. 
  19. [19] J. Crisp & L. Paoluzzi – « On the classification of CAT(0) structures for the 4-string braid group », Michigan Math. J.53 (2005), p. 133–163. Zbl1135.20030MR2125539
  20. [20] P. Deligne – « Les immeubles des groupes de tresses généralisés », Invent. Math.17 (1972), p. 273–302. Zbl0238.20034MR422673
  21. [21] J. L. Dyer & E. K. Grossman – « The automorphism groups of the braid groups », Amer. J. Math.103 (1981), p. 1151–1169. Zbl0476.20026MR636956
  22. [22] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson & W. P. Thurston – Word processing in groups, Jones and Bartlett Publishers, 1992. Zbl0764.20017MR1161694
  23. [23] F. A. Garside – « The braid group and other groups », Quart. J. Math. Oxford Ser.20 (1969), p. 235–254. Zbl0194.03303MR248801
  24. [24] R. Grigorchuk & T. Nagnibeda – « Complete growth functions of hyperbolic groups », Invent. Math.130 (1997), p. 159–188. Zbl0880.20024MR1471889
  25. [25] U. Haagerup – « An example of a nonnuclear C * -algebra, which has the metric approximation property », Invent. Math. 50 (1978/79), p. 279–293. Zbl0408.46046MR520930
  26. [26] P. de la Harpe – « Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint », C. R. Acad. Sci. Paris Sér. I Math.307 (1988), p. 771–774. Zbl0653.46059MR972078
  27. [27] P. Jolissaint – « Rapidly decreasing functions in reduced C * -algebras of groups », Trans. Amer. Math. Soc.317 (1990), p. 167–196. Zbl0711.46054MR943303
  28. [28] D. Krammer – « The braid group B 4 is linear », Invent. Math.142 (2000), p. 451–486. Zbl0988.20023MR1804157
  29. [29] V. Lafforgue – « A proof of property (RD) for cocompact lattices of SL ( 3 , 𝐑 ) and SL ( 3 , 𝐂 ) », J. Lie Theory10 (2000), p. 255–267. Zbl0981.46046MR1774859
  30. [30] —, « K -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes », Invent. Math.149 (2002), p. 1–95. Zbl1084.19003MR1914617
  31. [31] H. Oyono-Oyono – « Baum-Connes conjecture and group actions on trees », K -Theory 24 (2001), p. 115–134. Zbl1008.19001MR1869625
  32. [32] A. Piggott, K. Ruane & G. Walsh – « The automorphism group of the free group of rank two is a CAT(0) group », preprint arXiv:0809.2034. Zbl1205.20050MR2677622
  33. [33] J. Ramagge, G. Robertson & T. Steger – « A Haagerup inequality for A ˜ 1 × A ˜ 1 and A ˜ 2 buildings », Geom. Funct. Anal.8 (1998), p. 702–731. Zbl0906.43009MR1633983
  34. [34] M. Ronan – Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press Inc., 1989. Zbl0694.51001MR1005533
  35. [35] T. Schick – « Finite group extensions and the Baum-Connes conjecture », Geom. Topol.11 (2007), p. 1767–1775. Zbl1201.58019MR2350467
  36. [36] J. Tits – Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., vol. 386, Springer, 1974. Zbl0295.20047MR470099
  37. [37] A. Valette – Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser, 2002. Zbl1136.58013MR1907596

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.