Counting Growth Types of Automorphisms of Free Groups

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e, d, R) may be realized by an automorphism of Fn. In particular, the inequality e ￡ \frac3n-24{{e \leq \frac{3n-2}{4}}} (due to Levitt–Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism.