Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

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Infinite Abelian Groups

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

This paper compiles basic components of the construction of random groups and of the proof of their properties announced in [G10]. Justification of each step, as well as the interrelation between them, is straightforward by available techniques specific to each step. On the other hand, there are several ingredients that cannot be truly appreciated without extending the present framework. We shall indicate along the way possible developments, postponing full exposition to forthcoming articles expanding the following points touched upon in the present paper.

Random walk in random groups

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Metric Spaces Of Non Positive Curvature

Reducts of the random graph

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-free abelian groups of finite rank n > 2. So it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n > 2. Of course, if we wish to show that the classification problem for the groups of rank n > 2 is intractible, it is not enough merely to prove that there are 2' such groups up to isomorphism. For there are 2' pairwise nonisomorphic groups of rank 1, and we have already pointed out that Baer has given a satisfactory classification for this class of groups. In this paper, following Friedman-Stanley [11] and Hjorth-Kechris [15], we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n > 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space Qf which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank 1 < r < n can be naturally identified with the set S(Qn) of all nontrivial additive subgroups of Qn. Notice that S(Qn) is a Borel subset of the Polish space ,p(Qn) of all subsets of Qn, and hence S(Qn) can be regarded as a standard Borel space; i.e. a Polish space equipped with its associated o-algebra of Borel subsets. (Here we are identifying p(Qn) with the space 2'Qn of all functions h: Qn {0, 1} equipped with the product topology.) Furthermore, the natural action of GLn(Q) on the vector space Qn induces a corresponding Borel action on S(Qn); and it is easily checked that if A, B C S(Qn), then A _V B iff there exists an element f E GLn(Q) such that p(A) = B. It follows that the isomorphism relation on S(Qn) is a countable Borel equivalence relation. (If X is a standard Borel space, then a Borel equivalence relation on X is an equivalence relation E C X2 which is a Borel subset of X2. The Borel equivalence relation E is said to be countable iff every E-equivalence class is countable.)

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The classification problem for torsion-free abelian groups of finite rank

Asymptotic Cones of Finitely Generated Groups

http://yoksis.bilkent.edu.tr/doi_getpdf/articles/10.1016-0168-0072(95)00061-5.pdf

Reducts of random hypergraphs

Let \(\mathbb{F}\) be a finite field, and let (ℙ, B) be a nontrivial 2-(n, k, 1)-design over \(\mathbb{F}\). Then each point α∈ℙ induces a (k−1)-spread Sα on ℙ/α. (ℙ, B) is said to be a geometric design if Sα is a geometric spread on ℙ/α for each α∈ℙ. In this paper, we prove that there are no geometric designs over any finite field \(\mathbb{F}\).

Simon Thomas

Papers

Reducts of the random graph

The classification problem for torsion-free abelian groups of finite rank

Asymptotic Cones of Finitely Generated Groups

Reducts of random hypergraphs

Designs and partial geometries over finite fields