Random groups arising as graph products

Charney, Ruth¹ ; Farber, Michael²

¹Department of Mathematics, Brandeis University, Waltham MA 02453, USA
²Warwick Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 979-995

Résumé

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdös and Rényi model of a random graph and find precise threshold functions for hyperbolicity (or relative hyperbolicity). We also study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as n →∞, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter p is constant and satisfies 0.2929 < p < 1.

Détail

DOI : 10.2140/agt.2012.12.979

Classification : 20P05, 20F36, 57M07
Keywords: random group, right angled Artin group, hyperbolic group, automorphisms of Artin groups

Bibliographie

[1] N Alon, J H Spencer, The probabilistic method, , John Wiley Sons (2008)

[2] E Babson, C Hoffman, M Kahle, The fundamental group of random 2–complexes, J. Amer. Math. Soc. 24 (2011) 1

[3] J Behrstock, C Druţu, L Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann. 344 (2009) 543

[4] B Bollobás, Random graphs, 73, Cambridge Univ. Press (2001)

[5] C Champetier, Propriétés statistiques des groupes de présentation finie, Adv. Math. 116 (1995) 197

[6] R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141

[7] R Charney, J Crisp, Relative hyperbolicity and Artin groups, Geom. Dedicata 129 (2007) 1

[8] R Charney, K Ruane, N Stambaugh, A Vijayan, The automorphism group of a graph product with no SIL, Illinois J. Math. 54 (2010) 249

[9] R Charney, K Vogtmann, Finiteness properties of automorphism groups of right-angled Artin groups, Bull. Lond. Math. Soc. 41 (2009) 94

[10] R Charney, K Vogtmann, Subgroups and quotients of automorphism groups of RAAGs, from: "Low-dimensional and symplectic topology", Proc. Sympos. Pure Math. 82, Amer. Math. Soc. (2011) 9

[11] L J Corredor, M A Gutierrez, A generating set for the automorphism group of the graph product of abelian groups

[12] A Costa, M Farber, Topology of random right angled Artin groups, J. Topol. Anal. 3 (2011) 69

[13] M W Davis, Buildings are CAT(0), from: "Geometry and cohomology in group theory (Durham, 1994)", London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 108

[14] M W Davis, The geometry and topology of Coxeter groups, 32, Princeton Univ. Press (2008)

[15] M B Day, Finiteness of outer automorphism groups of random right-angled Artin groups

[16] M B Day, Peak reduction and finite presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009) 817

[17] P Erdős, A Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960) 17

[18] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810

[19] M Farber, Topology of random linkages, Algebr. Geom. Topol. 8 (2008) 155

[20] M Farber, T Kappeler, Betti numbers of random manifolds, Homology, Homotopy Appl. 10 (2008) 205

[21] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, Vol. 2 (Sussex, 1991)", London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1

[22] M Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003) 73

[23] S Hermiller, J Meier, Algorithms and geometry for graph products of groups, J. Algebra 171 (1995) 230

[24] S Janson, T Łuczak, A Rucinski, Random graphs, , Wiley-Interscience (2000)

[25] I Kapovich, P E Schupp, Random quotients of the modular group are rigid and essentially incompressible, J. Reine Angew. Math. 628 (2009) 91

[26] I Kapovich, P Schupp, V Shpilrain, Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups, Pacific J. Math. 223 (2006) 113

[27] M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. 52 (1995) 318

[28] N Linial, R Meshulam, Homological connectivity of random 2–complexes, Combinatorica 26 (2006) 475

[29] J Meier, When is the graph product of hyperbolic groups hyperbolic?, Geom. Dedicata 61 (1996) 29

[30] R Meshulam, N Wallach, Homological connectivity of random k–dimensional complexes, Random Structures Algorithms 34 (2009) 408

[31] G Moussong, Hyperbolic Coxeter groups, PhD thesis, Ohio State University (1988)

[32] Y Ollivier, A January 2005 invitation to random groups, 10, Sociedade Brasileira de Matemática (2005)

[33] H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989) 34

[34] A Shalev, Probabilistic group theory and Fuchsian groups, from: "Infinite groups: geometric, combinatorial and dynamical aspects", Progr. Math. 248, Birkhäuser (2005) 363

[35] A Zuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003) 643