Introduction

Alfred Tarski in 1940 made three well-known conjectures concerning nonabelian free groups (see Section 2). There had been various partial solutions until complete positive solutions were presented during the past 15 years by Kharlampovich and Myasnikov (see [51]–[59]) and independently by Z. Sela (see [78]–[83]). In the Kharlampovich- Myasnikov approach the proof arose from a detailed study of fully residually free groups (called limit groups in Sela's approach), the development of algebraic geometry over free groups, and an elimination process involving solutions of equations over free groups based on work of Makhanin and Razborov (see [51]–[59]). These steps were mirrored, with somewhat different terminology, by Sela, who called his approach diophantine geometry over free groups.

The positive solution of the Tarski conjectures provides a straightforward proof of Magnus's theorem in surface groups which we present. This result was proved directly by J. Howie [46] and independently by O. Bogopolski [10]. We will present this proof in Section 4. This type of proof leads to several different types of questions.

1. • Which additional nontrivial free group results are true in surface groups but difficult to obtain directly?

2. • What first-order properties of nonabelian free groups are true beyond the class of elementary free groups?

After showing a proof of Magnus's Theorem based on the solution of the Tarski problems we give several examples of other free group results holding in surface groups. Using this technique we give a proof of a theorem of D. Lee on C-test words. We then consider and prove certain other results that hold in elementary free groups, in particular surface groups, including the retract theorem of Turner [86] and the property of conjugacy separability.

After this we turn to the second type of question and survey a large number of recent results. In particular we first consider groups satisfying certain quadratic properties that we call Lyndon properties and show that the class of groups satisfying these properties are closed under many amalgam constructions.

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.

In this paper we use the terminology and notation of [12] and [51]. One finds there the concepts and results concerning combinatorial descriptions of groups which are used (but not always further explained) below.

THE NIELSEN REDUCTION METHOD IN AMALGAMATED FREE PRODUCTS AND HNN GROUPS

One of the most important methods in the theory of free groups and some similar groups, is the Nielsen reduction method. If F is the free group with free generators a, b, … one can define a notion of free length L in F (relative to the generators a, b, …) and a certain lexicographical ordering. The Nielsen reduction method in F concerns Nielsen transformation from systems {gj}jϵJ to systems which are shorter with respect to the length L and ordering (cf. [21], [22], [54]).

If we apply this method to a finite system {x1, …, xm}, we arrive after a finite number of steps at a system in which no element and no inverse of an element can shorten another by more than half, and no two can shorten another to nothing - i. e. a system which possesses the Nielsen property with respect to L. Nielsen ([21], [22]) used the property for the proof that subgroups of free groups are free. To be sure, one cannot define a meaningful length function in every group (cf. [4], [8], [13]). A length satisfying certain ‘natural’ axioms exists essentially only in free products.