Cubulating random groups at density less than 1/6

Ollivier, Yann; Wise, Daniel

doi:10.1090/S0002-9947-2011-05197-4

Cubulating random groups at density less than $1/6$
HTML articles powered by AMS MathViewer

by Yann Ollivier and Daniel T. Wise PDF

Trans. Amer. Math. Soc. 363 (2011), 4701-4733 Request permission

Abstract:

We prove that random groups at density less than $\frac 16$ act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than $\frac 15$ have codimension-$1$ subgroups. In particular, Property $(T)$ fails to hold at density less than $\frac 15$.

Abstract. Nous prouvons que les groupes aléatoires en densité strictement inférieure à $\frac 16$ agissent librement et cocompactement sur un complexe cubique CAT(0). De plus en densité strictement inférieure à $\frac 15$, ils ont un sous-groupe de codimension $1$; en particulier, la propriété $(T)$ n’est pas vérifiée.

References

Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property ($T$), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.
M. Bożejko, T. Januszkiewicz, and R. J. Spatzier, Infinite Coxeter groups do not have Kazhdan’s property, J. Operator Theory 19 (1988), no. 1, 63–67. MR 950825
W. Ballmann and J. Świątkowski, On $L^2$-cohomology and property (T) for automorphism groups of polyhedral cell complexes, Geom. Funct. Anal. 7 (1997), no. 4, 615–645. MR 1465598, DOI 10.1007/s000390050022
Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR 1852148, DOI 10.1007/978-3-0348-8237-8
Indira Chatterji, Cornelia Druţu, and Frédéric Haglund. Median spaces and spaces with measured walls. Preprint.
Pierre-Alain Cherix, Florian Martin, and Alain Valette, Spaces with measured walls, the Haagerup property and property (T), Ergodic Theory Dynam. Systems 24 (2004), no. 6, 1895–1908. MR 2106770, DOI 10.1017/S0143385704000185
Indira Chatterji and Graham Niblo, From wall spaces to $\rm CAT(0)$ cube complexes, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 875–885. MR 2197811, DOI 10.1142/S0218196705002669
Pierre de la Harpe and Alain Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989), 158 (French, with English summary). With an appendix by M. Burger. MR 1023471
Étienne Ghys, Groupes aléatoires (d’après Misha Gromov,$\dots$), Astérisque 294 (2004), viii, 173–204 (French, with French summary). MR 2111644
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
C. H. Houghton, Ends of locally compact groups and their coset spaces, J. Austral. Math. Soc. 17 (1974), 274–284. Collection of articles dedicated to the memory of Hanna Neumann, VII. MR 0357679
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.
Chris Hruska and Daniel T. Wise. Axioms for finiteness of cubulations. Preprint, 2004.
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Jonathan P. McCammond and Daniel T. Wise, Fans and ladders in small cancellation theory, Proc. London Math. Soc. (3) 84 (2002), no. 3, 599–644. MR 1888425, DOI 10.1112/S0024611502013424
Bogdan Nica, Cubulating spaces with walls, Algebr. Geom. Topol. 4 (2004), 297–309. MR 2059193, DOI 10.2140/agt.2004.4.297
Graham Niblo and Lawrence Reeves, Groups acting on $\textrm {CAT}(0)$ cube complexes, Geom. Topol. 1 (1997), approx. 7 pp.}, issn=1465-3060, review= MR 1432323, doi=10.2140/gt.1997.1.1,
Graham A. Niblo and Martin A. Roller, Groups acting on cubes and Kazhdan’s property (T), Proc. Amer. Math. Soc. 126 (1998), no. 3, 693–699. MR 1459140, DOI 10.1090/S0002-9939-98-04463-3
G. A. Niblo and L. D. Reeves, Coxeter groups act on $\textrm {CAT}(0)$ cube complexes, J. Group Theory 6 (2003), no. 3, 399–413. MR 1983376, DOI 10.1515/jgth.2003.028
Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (2004), no. 3, 595–679. MR 2100673, DOI 10.1007/s00039-004-0470-y
Yann Ollivier, Cogrowth and spectral gap of generic groups, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 289–317 (English, with English and French summaries). MR 2141699
Yann Ollivier, A January 2005 invitation to random groups, Ensaios Matemáticos [Mathematical Surveys], vol. 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. MR 2205306
Yann Ollivier, Some small cancellation properties of random groups, Internat. J. Algebra Comput. 17 (2007), no. 1, 37–51. MR 2300404, DOI 10.1142/S021819670700338X
Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406, DOI 10.1112/plms/s3-71.3.585
Michah Sageev, Codimension-$1$ subgroups and splittings of groups, J. Algebra 189 (1997), no. 2, 377–389. MR 1438181, DOI 10.1006/jabr.1996.6884
Peter Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 179–198. MR 487104, DOI 10.1016/0022-4049(77)90051-2
Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. MR 2053602, DOI 10.1007/s00039-004-0454-y
A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643–670. MR 1995802, DOI 10.1007/s00039-003-0425-8