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].