Skip to main content Accessibility help
×
Home
  • Print publication year: 2015
  • Online publication date: July 2015

5 - Randomly generated groups

Summary

Abstract

We discuss some older and a few recent results related to randomly generated groups. Although most of them are of topological and geometric flavour the main aim of this work is to present them in combinatorial settings.

1 Introduction

For the last half of the century the theory of randomly generated discrete structures has established itself as a vital part of combinatorics. Random graphs and hypergraphs and, more generally, combinatorial, algebraic, and geometric structures generated randomly have been used widely not only to provide numerous examples of objects of exotic properties but also as the way of studying and understanding large non-random systems which often can be decomposed into a small number of pseudorandom parts (see, for instance, Tao [37]). However, until recently, in the theory of random structures as known to combinatorialists random groups have not appeared very frequently (one is tempted to say, sporadically) although Gromov's model of the random group has already been introduced in the early eighties. The main reason was, undoubtedly, the fact that the world of combinatorialists seemed to be quite distant from the land of geometers and topologists and, despite many efforts of a few distinguished mathematicians familiar with both territories, combinatorialists did not believe that one can get basic understanding of the subject without much effort. This landscape has dramatically changed over the last few years. Topological combinatorics (or combinatorial topology) has been developing rapidly; many new projects have been started and a substantial number of articles have been published; combinatorialists have started to use topological terminology and more and more topological works are using advanced combinatorial tools. The aim of this article is just to spread the news. So it is not exactly a survey or even an introduction to this quickly evolving area – the reader who looks for this type of work is referred to a somewhat old but still excellent survey of Ollivier ([34], see also [35]) and the recent paper of Kahle [22].

Related content

Powered by UNSILO
[1] S., Antoniuk, E., Friedgut, T., Łuczak, A sharp threshold for collapse of the random triangular group, arXiv:1403.3516.
[2] S., Antoniuk, T., Łuczak, T., Prytula, P., Przytycki, B., Zalewski, When a random triangular group is free?, in preparation.
[3] S., Antoniuk, T., Łuczak, J., Świątkowski, Collapse of random triangular groups: a closer look, Bull. Lond. Math. Soc. 46 (2014), 761–764.
[4] S., Antoniuk, T., Łuczak, J., Świątkowski, Random triangular groups at density 1/3, Compositio Mathematica 151 (2015), 167–178.
[5] L., Aronshtam, N., Linial, The threshold for collapsibility in random complexes, Random Structures & Algorithms, to appear.
[6] L., Aronshtam, N., Linial, When does the top homology of a random simplicial complex vanish?, Random Structures & Algorithms 46 (2015), 26–35.
[7] L., Aronshtam, N., Linial, T., Łuczak, R., Meshulam, Collapsibility and vanishing of top homology in random simplicial complexes, Discrete Comput. Geom. 49 (2013), 317–334.
[8] E., Babson, C., Hoffman, M., Kahle, The fundamental group of random 2-complexes, J. Amer. Math. Soc. 24 (2011), 1–28.
[9] S. R., Blackburn, P. M., Neumann, G., Venkataraman, “Enumeration of finite groups”. Cambridge Tracts in Mathematics, 173. Cambridge University Press, Cambridge, 2007.
[10] B., Bollobás, The evolution of random graphs, Trans. Amer. Math. Soc. 286 (1984), 257–274.
[11] B., Bollobás, A., Thomason, Threshold functions, Combinatorica 7 (1987), 35–38.
[12] P.J., Cameron, The random graph revisited, European Congress of Mathematics, Vol. I (Barcelona, 2000), 267–274, Progr. Math., 201, Birkhäuser, Basel, 2001.
[13] B., DeMarco, A., Hamm, J., Kahn, On the triangle space of a random graph, J. Combin. 4 (2013), 229–249.
[14] J. D., Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205.
[15] P., Erdős, A., Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.
[16] E., Friedgut, Sharp thresholds of graph properties, and the k-sat problem. With an appendix by Jean Bourgain. J. Amer. Math. Soc. 12 (1999), 1017–1054.
[17] G., Grimmett, S., Janson; Random even graphs. Electron. J. Combin. 16 (2009), no. 1, Research Paper, 46, 19 pp.
[18] M., Gromov, Asymptotic invariants of infinite groups. Geometric Group Theory, London Math. Soc. Lecture Note Ser. 182 (1993), 1–295.
[19] P., Heinig, T., Łuczak, Hamiltonian space of random graphs, in preparation.
[20] C., Hoffman, M., Kahle, E., Paquette, Spectral gaps of random graphs and applications to random topology, arXiv:1201.0425.
[21] S., Janson, T., Łuczak, A., Ruciński, “Random Graphs”, Wiley, New York, 2000.
[22] M., Kahle, Topology of random simplicial complexes: a survey. To appear in AMS Contemporary Volumes in Mathematics, Nov 2014, arXiv:1301.7165.
[23] M., Kotowski, M., Kotowski, Random groups and property (T): Żuk's theorem revisited, J. London Math. Soc. 88 (2013), 396–416.
[24] N., Linial, R., Meshulam, Homological connectivity of random 2- complexes, Combinatorica 26 (2006), 475–487.
[25] N., Linial, Y., Peled, On the phase transition in random simplicial complexes, arXiv:1410.1281.
[26] T., Łuczak, The automorphisms group of random graphs with given number of edges, Math. Proc. Camb. Phil. Soc. 104 (1988), 441–449.
[27] T., Łuczak, Components behavior near the critical point of the random graph process, Random Structures … Algorithms 1 (1990), 287–310.
[28] T., Łuczak, Cycles in a random graph near the critical point, Random Structures & Algorithms 2 (1991), 421–440.
[29] T., Łuczak, Size and connectivity of the k-core of a random graph, Discrete Math. 91 (1991), 61–68.
[30] T., Łuczak, How to deal with unlabelled random graphs, J. Graph Theory 15 (1991), 303–316.
[31] T., Łuczak, L., Pyber, On random generation of the symmetric group, Combinatorics, Probability & Computing 2 (1993), 505–512.
[32] R., Meshulam, N., Wallach, Homological connectivity of random kdimensional complexes, Random Structures & Algorithms 34 (2009), 408–417.
[33] T., OdrzygóźdźThe square model for random groups, arXiv:1405.2773.
[34] Y., Ollivier, A January 2005 invitation to random groups, Ensaios Matematicos [Mathematical Surveys] 10, Sociedade Brasileira de Matematica, Rio de Janeiro, 2005.
[35] Y., Ollivier, Random group update, http://www.yannollivier.org/rech/publs/rgupdates.pdf.
[36] Y., Ollivier, D. T., Wise, Cubulating random groups at density less than 1/6, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4701–4733.
[37] T., Tao, The dichotomy between structure and randomness, arithmetic progressions, and the primes, International Congress of Mathematicians. Vol. I, 581–608, Eur. Math. Soc., Zürich, 2007.
[38] A., Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), 643–670.