Elementary theory of groups

Summary

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,…, x_n, g₁,…, g_k) in free variables x₁,…, x_n and constants g₁,…, g_k which are elements of G. A solution consists of an n-tuple (h₁,…, h_n) of elements from G which upon substitution for x₁,…, x_n 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.