Book contents
- Frontmatter
- Contents
- Preface
- 1 Ramsey classes: examples and constructions
- 2 Recent developments in graph Ramsey theory
- 3 Controllability and matchings in random bipartite graphs
- 4 Some old and new problems in combinatorial geometry I: around Borsuk's problem
- 5 Randomly generated groups
- 6 Curves over finite fields and linear recurring sequences
- 7 New tools and results in graph minor structure theory
- 8 Well quasi-order in combinatorics: embeddings and homomorphisms
- 9 Constructions of block codes from algebraic curves over finite fields
- References
5 - Randomly generated groups
Published online by Cambridge University Press: 05 July 2015
- Frontmatter
- Contents
- Preface
- 1 Ramsey classes: examples and constructions
- 2 Recent developments in graph Ramsey theory
- 3 Controllability and matchings in random bipartite graphs
- 4 Some old and new problems in combinatorial geometry I: around Borsuk's problem
- 5 Randomly generated groups
- 6 Curves over finite fields and linear recurring sequences
- 7 New tools and results in graph minor structure theory
- 8 Well quasi-order in combinatorics: embeddings and homomorphisms
- 9 Constructions of block codes from algebraic curves over finite fields
- References
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].
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- Surveys in Combinatorics 2015 , pp. 175 - 194Publisher: Cambridge University PressPrint publication year: 2015