Growth and displacement of free product automorphisms

TL;DR

This paper extends the understanding of automorphism growth rates from irreducible free groups to general and free product cases, linking growth to minimal displacement in a broader context.

Contribution

It generalizes the relationship between growth rate and minimal displacement to reducible automorphisms and free products, beyond the irreducible case.

Findings

Growth rate equals minimal displacement for reducible automorphisms

Extension of train track techniques to free product automorphisms

Introduction of relative growth concept for free products

Abstract

It is well known for an irreducible free group automorphism that its growth rate is equal to the minimal Lipschitz displacement of its action on Culler-Vogtmann space. This follows as a consequence of the existence of train track representatives for the automorphism. We extend this result to the general - possibly reducible - case as well as to the free product situation where growth is replaced by `relative growth'.