Folding free-group automorphisms

TL;DR

This paper presents an algorithm leveraging Stallings' folding technique to decompose automorphisms of free groups into Whitehead automorphisms, providing new methods for subgroup generation and intersection analysis.

Contribution

It introduces an algorithm for decomposing automorphisms of free groups and offers new approaches to generating subgroups and analyzing their intersections.

Findings

Algorithm effectively decomposes automorphisms into Whitehead automorphisms.

Provides finite generating sets for specific subgroups of automorphisms.

Shows the intersection of certain subgroups with IA_n is finitely generated.

Abstract

We describe an algorithm that uses Stallings' folding technique to decompose an element of $Aut(F_n)$ as a product of Whitehead automorphisms (and hence as a product of Nielsen transformations.) We use this to give an alternative method of finding a finite generating set for the subgroup of $Aut(F_n)$ that fixes a subset $Y$ of the basis elements, and the subgroup that fixes each element of $Y$ up to conjugacy. We show that the intersection of this latter subgroup with $IA_n$ is also finitely generated.