Sur le groupe d'automorphismes du groupe libre I. Transvections

Abstract

In this study we define a new family of free automorphisms called transvections which generalise the Whitehead automorphisms. This leads to a new presentation of the group of free automorphisms and to a partial generalisation of McCool's well known result on the presentation of certain of its subgroups. We study the “linear” characteristics of transvections with the use of Fox free differential calculus. The groups generated by these automorphisms form the family of transvection groups that one could name the pre-classical groups. We give some simple examples of these groups, which we shall study further throughout the following papers.