Cubulating random groups at density less than \(1/6\). (English) Zbl 1277.20048
From the introduction: We prove that random groups at density less than \(\tfrac16\) act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than \(\tfrac15\) have codimension-1 subgroups. In particular, Property (T) fails to hold at density less than \(\tfrac15\).
M. Gromov introduced [in Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Lond. Math. Soc. Lect. Note Ser. 182. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)] the notion of a random finitely presented group on \(m\geq 2\) generators at density \(d\in (0;1)\). The idea is to fix a set \(\{g_1,\dots,g_m\}\) of generators and to consider presentations with \((2m-1)^{d\ell}\) relations each of which is a random reduced word of length \(\ell\) (Definition 1.1). The ‘density’ \(d\) is a measure of the size of the number of relations as compared to the total number of available relations. See Section 1 for precise definitions and basic properties.
One of the striking facts Gromov proved is that a random finitely presented group is infinite, hyperbolic at density \(<\tfrac12\), and is trivial or \(\{\pm 1\}\) at density \(>\tfrac12\), with probability tending to 1 as \(\ell\to\infty\).
As above, the statements about the behavior of a group at a certain density are only correct with probability tending to 1 as \(\ell\to\infty\). Throughout the paper, we will say that a given property holds ‘with overwhelming probability’ if its probability tends exponentially to 1 as \(\ell\to\infty\).
The goals of this paper are a complete geometrization theorem at \(d<\tfrac16\), implying the Haagerup property, and existence of a codimension-1 subgroup at \(d<\tfrac15\), implying [G. A. Niblo, M. A. Roller, Proc. Am. Math. Soc. 126, No. 3, 693-699 (1998; Zbl 0906.20024)] failure of Property (T):
Theorem 10.4. With overwhelming probability, random groups at density \(d<\tfrac16\) act freely and cocompactly on a CAT(0) cube complex.
Corollary 9.2. With overwhelming probability, random groups at density \(d<\tfrac16\) are a-T-menable (Haagerup property).
Theorem 7.4. With overwhelming probability, random groups \(G\) at density \(d<\tfrac15\) have a subgroup \(H\) which is free, quasiconvex and such that the relative number of ends \(e(G,H)\) is at least 2.
Corollary 7.5. With overwhelming probability, random groups at density \(d<\tfrac15\) do not have Property (T).
M. Gromov introduced [in Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Lond. Math. Soc. Lect. Note Ser. 182. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)] the notion of a random finitely presented group on \(m\geq 2\) generators at density \(d\in (0;1)\). The idea is to fix a set \(\{g_1,\dots,g_m\}\) of generators and to consider presentations with \((2m-1)^{d\ell}\) relations each of which is a random reduced word of length \(\ell\) (Definition 1.1). The ‘density’ \(d\) is a measure of the size of the number of relations as compared to the total number of available relations. See Section 1 for precise definitions and basic properties.
One of the striking facts Gromov proved is that a random finitely presented group is infinite, hyperbolic at density \(<\tfrac12\), and is trivial or \(\{\pm 1\}\) at density \(>\tfrac12\), with probability tending to 1 as \(\ell\to\infty\).
As above, the statements about the behavior of a group at a certain density are only correct with probability tending to 1 as \(\ell\to\infty\). Throughout the paper, we will say that a given property holds ‘with overwhelming probability’ if its probability tends exponentially to 1 as \(\ell\to\infty\).
The goals of this paper are a complete geometrization theorem at \(d<\tfrac16\), implying the Haagerup property, and existence of a codimension-1 subgroup at \(d<\tfrac15\), implying [G. A. Niblo, M. A. Roller, Proc. Am. Math. Soc. 126, No. 3, 693-699 (1998; Zbl 0906.20024)] failure of Property (T):
Theorem 10.4. With overwhelming probability, random groups at density \(d<\tfrac16\) act freely and cocompactly on a CAT(0) cube complex.
Corollary 9.2. With overwhelming probability, random groups at density \(d<\tfrac16\) are a-T-menable (Haagerup property).
Theorem 7.4. With overwhelming probability, random groups \(G\) at density \(d<\tfrac15\) have a subgroup \(H\) which is free, quasiconvex and such that the relative number of ends \(e(G,H)\) is at least 2.
Corollary 7.5. With overwhelming probability, random groups at density \(d<\tfrac15\) do not have Property (T).
MSC:
| 20F65 | Geometric group theory |
| 20P05 | Probabilistic methods in group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20F69 | Asymptotic properties of groups |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
CAT(0) cube complexes; random groups; random finitely presented groups; random reduced words; property (T)References:
| [1] | Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (\( T\)), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008. · Zbl 1146.22009 |
| [2] | 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. · Zbl 0662.20040 |
| [3] | W. Ballmann and J. Świątkowski, On \?²-cohomology and property (T) for automorphism groups of polyhedral cell complexes, Geom. Funct. Anal. 7 (1997), no. 4, 615 – 645. · Zbl 0897.22007 · doi:10.1007/s000390050022 |
| [4] | 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. · Zbl 1030.43002 |
| [5] | Indira Chatterji, Cornelia Druţu, and Frédéric Haglund. Median spaces and spaces with measured walls. Preprint. |
| [6] | 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. · Zbl 1068.43007 · doi:10.1017/S0143385704000185 |
| [7] | Indira Chatterji and Graham Niblo, From wall spaces to \?\?\?(0) cube complexes, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 875 – 885. · Zbl 1107.20027 · doi:10.1142/S0218196705002669 |
| [8] | Pierre de la Harpe and Alain Valette, La propriété (\?) 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. · Zbl 0759.22001 |
| [9] | Étienne Ghys, Groupes aléatoires (d’après Misha Gromov,…), Astérisque 294 (2004), viii, 173 – 204 (French, with French summary). · Zbl 1134.20306 |
| [10] | 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. · Zbl 0841.20039 |
| [11] | 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. · Zbl 0289.22005 |
| [12] | 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. |
| [13] | Chris Hruska and Daniel T. Wise. Axioms for finiteness of cubulations. Preprint, 2004. |
| [14] | Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023 |
| [15] | 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. · Zbl 1022.20012 · doi:10.1112/S0024611502013424 |
| [16] | Bogdan Nica, Cubulating spaces with walls, Algebr. Geom. Topol. 4 (2004), 297 – 309. · Zbl 1131.20030 · doi:10.2140/agt.2004.4.297 |
| [17] | Graham Niblo and Lawrence Reeves, Groups acting on \?\?\?(0) cube complexes, Geom. Topol. 1 (1997), approx. 7 pp., issn=1465-3060, review=\MR{1432323}, doi=10.2140/gt.1997.1.1,. · Zbl 0887.20016 |
| [18] | 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. · Zbl 0906.20024 |
| [19] | G. A. Niblo and L. D. Reeves, Coxeter groups act on \?\?\?(0) cube complexes, J. Group Theory 6 (2003), no. 3, 399 – 413. · Zbl 1068.20040 · doi:10.1515/jgth.2003.028 |
| [20] | Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (2004), no. 3, 595 – 679. · Zbl 1064.20045 · doi:10.1007/s00039-004-0470-y |
| [21] | 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). · Zbl 1133.20032 |
| [22] | Yann Ollivier, A January 2005 invitation to random groups, Ensaios Matemáticos [Mathematical Surveys], vol. 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. · Zbl 1163.20311 |
| [23] | Yann Ollivier, Some small cancellation properties of random groups, Internat. J. Algebra Comput. 17 (2007), no. 1, 37 – 51. · Zbl 1173.20025 · doi:10.1142/S021819670700338X |
| [24] | Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585 – 617. · Zbl 0861.20041 · doi:10.1112/plms/s3-71.3.585 |
| [25] | Michah Sageev, Codimension-1 subgroups and splittings of groups, J. Algebra 189 (1997), no. 2, 377 – 389. · Zbl 0873.20028 · doi:10.1006/jabr.1996.6884 |
| [26] | Peter Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 179 – 198. · Zbl 0368.20021 · doi:10.1016/0022-4049(77)90051-2 |
| [27] | 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. · Zbl 1013.20001 |
| [28] | D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150 – 214. · Zbl 1071.20038 · doi:10.1007/s00039-004-0454-y |
| [29] | A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643 – 670. · Zbl 1036.22004 · doi:10.1007/s00039-003-0425-8 |
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