Detecting the Growth of Free Group Automorphisms by Their Action on the Homology of Subgroups of Finite Index

In this paper we prove that if F is a finitely generated free group and φ ∈ Aut(F) is a polynomially growing automorphism then there exists a characteristic subgroup S ≤ F of finite index such that the automorphism of S induced by φ grows polynomially of the same degree as φ. The proof is geometric in nature and makes use of Improved Relative Train Track representatives of free group automorphisms. The study of automorphisms of non-abelian free groups has been reinvigorated in recent years by a program to understand free group automorphisms as homotopy equivalences of finite graphs, called topological representatives (see, for example, [5, 3, 2, 1]). This programme is driven largely by analogy with the study of surface automorphisms and has led to significant progress in the field. In a series of papers, Bestvina, Feighn and Handel have developed powerful normal forms for topological representatives, called Improved Relative Train Track (IRTT) representatives (in analogy with train track representatives of surface automorphisms) [3, 2, 1]. This technology has allowed them to prove a number of important results, most notably the Scott Conjecture [3] and the Tits Alternative for Out(F) [2, 1]. In many applications, such as our Main Theorem, the detailed structure inherent in IRTT representatives allows one to use geometric intuition to evade difficult and unsightly cancellation arguments. Let F be a finitely generated non-abelian free group and φ ∈ Aut(F) an automorphism. The growth function Gφ : N → N of φ quantifies the rate at which repeated application of the automorphism changes the ‘size’ of a basis of F (see §1). The asymptotic behaviour of Gφ does not depend on the choice of basis for F and is robust when passing to subgroups of finite index. We write F ab for the abelianisation of F ; Date: 28 July, 2004.