Lifting automorphisms: a survey

Summary

Introduction

Let F = F_n be the free group of a finite rank n ≥ 2 with a fixed set {x_i;, 1 ≤ i ≤ n} of free generators. If R is a characteristic subgroup of the group F then the natural homomorphism ∈_R : F → F/R induces the mapping τ_R : AutF → Aut(F/R) of the corresponding automorphism groups. Those automorphisms of the group F/R that belong to the image of τ_R are usually called tame. In this survey, we will be concerned with the following general question: How to determine whether or not a given automorphism of a group F/R is tame? In a more general situation, when R is an arbitrary normal subgroup of F, one can ask if a given generating system of the group F/R can be lifted to a generating system of F (in this case the system will be also called tame). This question has important applications to low-dimensional topology (see, for instance, [LuMol]).

The questions of lifting automorphisms and generating systems naturally give rise to the following two problems of independent interest:

1. (1) Finding appropriate necessary and/or sufficient condition(s) for an endomorphism of the group F_n to be an automorphism;

2. (2) Describing (in one or another way) the group Ant(F/R) or generating systems of a group F/R.