Isoperimetric inequalities for automorphism groups of free groups

In this paper we show that the groups of automorphisms and outer automorphisms of a finitely generated free group have isoperimetric functions which are bounded above by exponential functions. This exponential bound is best possible if the free group has rank three, but the best bound remains unknown in higher rank. Our techniques show more generally that n-dimensional isoperimetric functions for these groups are at most exponential for all n. A variation of the technique gives an asynchronous bounded combing of the mapping class group of a bounded surface.