Counting growth types of automorphisms of free groups

TL;DR

This paper characterizes the possible combinations of exponential growth, polynomial degree, and fixed subgroup rank for automorphisms of free groups, establishing bounds and realizing all feasible triples.

Contribution

It precisely determines which triples (e, d, R) can be realized by automorphisms of free groups, including confirming the Levitt-Lustig inequality.

Findings

The inequality e ≤ (3n-2)/4 always holds.

All feasible triples (e, d, R) are realizable.

Conjugacy class growth is polynomial times exponential.

Abstract

Given an automorphism of a free group $F_n$, 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 $F_n$. In particular, the inequality $e\le (3n-2)/4}$ (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.