Automorphisms of one-relator groups

Abstract

It is a well known fact that every group G has a presentation of the form G=F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut (G), we consider the subgroup Stab (R) ⊆ Aut (F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) → Aut (G) is onto; if it is, we can try to determine its kernel.

Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the more difficult part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups and determine the kernel of the above-mentioned homomorphism for a rather general class of those groups.