6 - GL_n and the automorphism groups of free metabelian groups and polynomial rings

Summary

INTRODUCTION

For an integer n ≥ 2, we consider the following three groups:

1. GL_n-1(R), the group of invertible (n - 1) × (n - 1) matrices with entries in R = ℤ [x₁, …,x_m] or K[x₁, …,x_m+1], m ≥ 1, the polynomial ring over the integers ≥ or a field K.

2. Aut(M(n)), the group of automorphisms of the free metabelian groups M(n) = F(n)/F″(n) of rank n where F(n) is the free group of rank n and F″(n) its second derived group.

3. Aut_K(P(n)), the group of K-automorphisms of the polynomial ring P(n) = K[x₁,…,x_n] in n indeterminates over the field K.

4. Aut_K(P(n)), the group of K-automorphisms of the polynomial ring P(n) = K[x₁,…,x_n] in n indeterminates over the field K.

A study of these groups has split into three distinct phases corresponding to values n = 2, n = 3 and n ≥ 4. Our theme is that the three groups have many similar or analogous characteristics and should be considered as one and the same study.

Consider for example an immediate question confronting one in a study of these groups. What is a reasonable set of generators? More specifically, do any of these groups possess elements other than the obvious or tame ones? (See §2 for definitions). Apart from the case n = 3, evidence is pointing to the fact that these groups possess only tame elements. The exceptional case n = 3 is indeed very different from the cases n ≠ 3, as we shall see, but apparently similar for all three groups. We attempt to justify this apparent similarity in our explanations below.