When is a polynomially growing automorphism of $F_n$ geometric ?

TL;DR

This paper provides an algorithmic method to determine when a polynomially growing automorphism of a free group is geometric, by representing it as an iterated Dehn twist and applying existing algorithms and classical results.

Contribution

It introduces an algorithm to decide geometricity of polynomially growing automorphisms of free groups, using Dehn twist representations and prior theoretical tools.

Findings

Algorithmic criterion for geometricity of automorphisms.

Representation of polynomially growing automorphisms as iterated Dehn twists.

Utilization of Whitehead algorithm and previous results to prove the main claim.

Abstract

The main result of this paper is an algorithmic answer to the question raised in the title, up to replacing the given $\hat{\phi} \in Out(F_n)$ by a positive power. In order to provide this algorithm, it is shown that every polynomially growing automorphism $\hat \phi$ can be represented by an iterated Dehn twist on some graph-of-groups $\cal{G}$ with $\pi_1{\cal{G}} = F_n$. One then uses results of two previous papers \cite{KY01, KY02} as well as some classical results such as the Whitehead algorithm to prove the claim.