Abstract

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by Kharlampovich and Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

Introduction

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by O. Kharlampovich and A. Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

In Section 1, we define discrimination for groups and describe its ties to other areas. Also the concept of trivially discriminating (TD) groups is introduced, and the concept of squarelike groups is defined. It is also indicated how to define discrimination for arbitrary algebraic systems. It is also shown how to generalize the concept of squarelike to arbitrary algebras.

Abstract

Remarkable ties between group theory, logic and algebraic geometry have come to light via the positive solution of the Tarski conjecture. This has led to further work on the universal theory of groups. In this paper we describe and survey this material a large body of which is not familiar to most group theorists

Contents

1. Introduction.

2. First Order Languages and Model Theory.

3. The Tarksi Problems.

4. Residually Free and Universally Free Groups.

5. Algebraic Geometry over Groups and Applications.

6. The Positive Solution to the Tarski Problems.

7. Discriminating, Co-discriminating and Squarelike Groups.

8. Open Questions.

Introduction

The elementary theory of a group G consists of all the first-order or elementary sentences (see section 2) which are true in G. Although this is a concept which originated in formal logic, in particular model theory, it arises independently from the theory of equations within groups. Recall that an equation in a group G is a word W(x1,…, xn, g1,…, gk) in free variables x1,…, xn and constants g1,…, gk which are elements of G. A solution consists of an n-tuple (h1,…, hn) of elements from G which upon substitution for x1,…, xn make the word trivial in G. Hence an equation is a first-order sentence in the language L[G] consisting of the elementary language of group theory (again see section 2) augmented by allowing constants from the group G.

Introduction

In a previous paper [1], the quotient groups of the lower central series Ḡn = Gn/Gn+1 were studied. There the group G was assumed to be a free product of a finite number of finitely generated Abelian groups and Gn denoted the nth subgroup of the lower central series of G. Here we give an improved proof of a complicated lemma which first appeared in [1] (in particular, Lemma 4.4 of [1]). The proof given here, especially for property (iii) of the conclusion of that lemma, is a significant simplification of that which appears in [1]. We observe that one of the consequences of Lemma 4.4 of [1] is to give a set of free generators for the lower central quotients in the case where the free factors are torsion free (i.e., G = J in the terminology of [1]. Moreover the free generators are the J-basic commutators also using the terminology of Definition 4.1 of [1]). The authors only thought of this simplification after the publication of [1]. Furthermore, our improved proof uses results from [2] and [3].

In this paper, we will employ the notation, terminology, results, references, and equations of [1]. Furthermore, the numbering, a.b, of any definition, equation, etc., of [1] will correspond here to a.b-I. For example, Lemma 2.1-I will mean Lemma 2.1 of [1].

Preliminaries

In order to carry out our goal, we need to give some preliminary machinery.

This volume contains the Annual High School Mathematics Examinations, given 1973 through 1982. It is a continuation of Contest Problem Books I, II, III, published as Volumes 5, 17, and 25 of the New Mathematical Library series and which contain the first twenty-three annual examinations. The Annual High School Mathematics Examinations (AHSME), it is hoped, provide challenging problems which teach, stimulate and provide enjoyment for not only the participants, but also the readers of these volumes.

All high school students are eligible to participate in the Annual High School Mathematics Examinations. In 1982, over 418,000 students in the United States, Canada, Puerto Rico, Colombia, Jamaica, Australia, Italy, England, Hungary, Ireland, Israel, Finland, Belgium and Luxembourg participated in the examination. It was administered also in many APO/FPO and other schools abroad. Each year a Summary of Results and Awards is sent to all participating high schools (in the United States and Canada). The problems are designed so that they can be solved with a knowledge of only “pre-calculus” mathematics, with emphasis on intermediate algebra and plane geometry. The subject classification at the end of the volume indicates which questions are related to which topics.

The problems on each examination become progressively more difficult, Between 1973 and 1977, the participants were given eighty minutes to complete the examination, and in subsequent years they were allowed ninety minutes.