Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cubulating random groups at density less than $ 1/6$

Authors: Yann Ollivier and Daniel T. Wise
Journal: Trans. Amer. Math. Soc. 363 (2011), 4701-4733
MSC (2010): Primary 20P05, 20F67; Secondary 20F65, 20F05, 20F06
Published electronically: March 28, 2011
MathSciNet review: 2806688
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that random groups at density less than $ \frac16$ act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than $ \frac15$ have codimension-$ 1$ subgroups. In particular, Property $ (T)$ fails to hold at density less than $ \frac15$.

ABSTRACT. Nous prouvons que les groupes aléatoires en densité strictement inférieure à $ \frac16$ agissent librement et cocompactement sur un complexe cubique CAT(0). De plus en densité strictement inférieure à $ \frac15$, ils ont un sous-groupe de codimension $ 1$; en particulier, la propriété $ (T)$ n'est pas vérifiée.

References [Enhancements On Off] (What's this?)

  • [BdlHV08] Bachir Bekka, Pierre de la Harpe, and Alain Valette.
    Kazhdan's property ($ T$), volume 11 of New Mathematical Monographs.
    Cambridge University Press, Cambridge, 2008.
  • [BJS88] M. Bożejko, T. Januszkiewicz, and R. J. Spatzier.
    Infinite Coxeter groups do not have Kazhdan's property.
    J. Operator Theory, 19(1):63-67, 1988. MR 950825 (89i:22025)
  • [BŚ97] W. Ballmann and J. Świ atkowski.
    On $ {L}\sp 2$-cohomology and property (T) for automorphism groups of polyhedral cell complexes.
    Geom. Funct. Anal., 7(4):615-645, 1997. MR 1465598 (98m:20043)
  • [CCJ$ ^+$01] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette. Groups with the Haagerup property, Gromov's a-T-menability. volume 197 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2001. MR 1852148 (2002h:22007)
  • [CDH] Indira Chatterji, Cornelia Druţu, and Frédéric Haglund.
    Median spaces and spaces with measured walls.
  • [CMV04] Pierre-Alain Cherix, Florian Martin, and Alain Valette.
    Spaces with measured walls, the Haagerup property and property (T).
    Ergodic Theory Dynam. Systems, 24(6):1895-1908, 2004. MR 2106770 (2005i:22006)
  • [CN05] Indira Chatterji and Graham Niblo.
    From wall spaces to $ \operatorname{CAT}(0)$ cube complexes.
    Internat. J. Algebra Comput., 15(5-6):875-885, 2005. MR 2197811 (2006m:20064)
  • [dlHV89] Pierre de la Harpe and Alain Valette.
    La propriété $ (T)$ de Kazhdan pour les groupes localement compacts.
    Number 175 in Astérisque. Soc. Math. France, 1989.
    With an appendix by M. Burger. MR 1023471 (90m:22001)
  • [Ghy04] Étienne Ghys.
    Groupes aléatoires (d'après Misha Gromov,$ \dots$).
    Astérisque, (294):173-204, 2004.
    Séminaire Bourbaki, Vol. 2003/2004, Exp. 916. MR 2111644 (2005j:20049)
  • [Gro93] M. Gromov.
    Asymptotic invariants of infinite groups.
    In Geometric group theory, Vol. 2 (Sussex, 1991), pages 1-295. Cambridge Univ. Press, Cambridge, 1993. MR 1253544 (95m:20041)
  • [Hou74] C. H. Houghton.
    Ends of locally compact groups and their coset spaces.
    J. Austral. Math. Soc., 17:274-284, 1974.
    Collection of articles dedicated to the memory of Hanna Neumann, VII. MR 0357679 (50:10147)
  • [HP98] Frédéric Haglund and Frédéric Paulin.
    Simplicité de groupes d'automorphismes d'espaces à courbure négative.
    In The Epstein birthday schrift, pages 181-248 (electronic). Geom. Topol., Coventry, 1998.
  • [HW04] Chris Hruska and Daniel T. Wise.
    Axioms for finiteness of cubulations.
    Preprint, 2004.
  • [LS77] Roger C. Lyndon and Paul E. Schupp.
    Combinatorial group theory.
    Springer-Verlag, Berlin, 1977.
    Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064 (58:28182)
  • [MW02] Jonathan P. McCammond and Daniel T. Wise.
    Fans and ladders in small cancellation theory.
    Proc. London Math. Soc. (3), 84(3):599-644, 2002. MR 1888425 (2003b:20047)
  • [Nic04] Bogdan Nica.
    Cubulating spaces with walls.
    Algebr. Geom. Topol., 4:297-309 (electronic), 2004. MR 2059193 (2005b:20076)
  • [NR97] Graham Niblo and Lawrence Reeves.
    Groups acting on $ {{\operatorname{CAT}}(0)}$ cube complexes.
    Geom. Topol., 1:approx. 7 pp. (electronic), 1997. MR 1432323 (98d:57005)
  • [NR98] Graham A. Niblo and Martin A. Roller.
    Groups acting on cubes and Kazhdan's property (T).
    Proc. Amer. Math. Soc., 126(3):693-699, 1998. MR 1459140 (98k:20058)
  • [NR03] G. A. Niblo and L. D. Reeves.
    Coxeter groups act on $ {\operatorname{CAT}}(0)$ cube complexes.
    J. Group Theory, 6(3):399-413, 2003. MR 1983376 (2004e:20072)
  • [Oll04] Y. Ollivier.
    Sharp phase transition theorems for hyperbolicity of random groups.
    Geom. Funct. Anal., 14(3):595-679, 2004. MR 2100673 (2005m:20101)
  • [Oll05a] Yann Ollivier.
    Cogrowth and spectral gap of generic groups.
    Ann. Inst. Fourier (Grenoble), 55(1):289-317, 2005. MR 2141699 (2006c:20134)
  • [Oll05b] Yann Ollivier.
    A January 2005 invitation to random groups, volume 10 of Ensaios Matemáticos [Mathematical Surveys].
    Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. MR 2205306 (2007e:20088)
  • [Oll07] Yann Ollivier.
    Some small cancellation properties of random groups.
    Internat. J. Algebra Comput., 17(1):37-51, 2007. MR 2300404 (2008g:20096)
  • [Sag95] Michah Sageev.
    Ends of group pairs and non-positively curved cube complexes.
    Proc. London Math. Soc. (3), 71(3):585-617, 1995. MR 1347406 (97a:20062)
  • [Sag97] Michah Sageev.
    Codimension-$ 1$ subgroups and splittings of groups.
    J. Algebra, 189(2):377-389, 1997. MR 1438181 (98c:20071)
  • [Sco78] Peter Scott.
    Ends of pairs of groups.
    J. Pure Appl. Algebra, 11(1-3):179-198, 1977/78. MR 487104 (81h:20047)
  • [Ser80] Jean-Pierre Serre.
    Springer-Verlag, Berlin, 1980.
    Translated from the French by John Stillwell. MR 1954121 (2003m:20032)
  • [Wis04] Daniel T. Wise.
    Cubulating small cancellation groups.
    Geom. Funct. Anal., 14(1):150-214, 2004. MR 2053602 (2005c:20069)
  • [Żuk03] A. Żuk.
    Property (T) and Kazhdan constants for discrete groups.
    Geom. Funct. Anal., 13(3):643-670, 2003. MR 1995802 (2004m:20079)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20P05, 20F67, 20F65, 20F05, 20F06

Retrieve articles in all journals with MSC (2010): 20P05, 20F67, 20F65, 20F05, 20F06

Additional Information

Yann Ollivier
Affiliation: CNRS, UMPA, École normale supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 7, France
Address at time of publication: CNRS, LRI, Université Paris-Sud, Bat. 490, 91405 Orsay cedex, France

Daniel T. Wise
Affiliation: Department of Mathematics, McGill University, Montreal, Québec, Canada H3A 2K6

Keywords: CAT(0) cube complexes, random groups, Property $(T)$
Received by editor(s): August 27, 2008
Received by editor(s) in revised form: August 27, 2009
Published electronically: March 28, 2011
Additional Notes: This research was partially supported by NSERC grant
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society