Partial Dehn twists of free groups relative to local Dehn twists - a dichotomy

TL;DR

This paper establishes a criterion to classify automorphisms of free groups based on their growth rate, showing that partial Dehn twists are either genuine Dehn twists or exhibit quadratic growth.

Contribution

It introduces a new criterion for automorphism growth classification and applies it to partial Dehn twists relative to local Dehn twists, revealing a clear dichotomy.

Findings

Partial Dehn twists are either honest Dehn twists or have quadratic growth.

The criterion effectively distinguishes growth types of automorphisms.

Application to free group automorphisms clarifies their structural behavior.

Abstract

A criterion for quadratic or higher growth of group automorphisms is established which are represented by graph-of-groups automorphisms with certain well specified properties. As a consequence, it is derived (using results of a previous paper of the author) that every partial Dehn twist automorphism of $\FN$ relative to local Dehn twist automorphisms is either an honest Dehn twist automorphism, or else has quadratic growth.