A McCool Whitehead type theorem for finitely generated subgroups of $\mathsf{Out}(F_n)$

TL;DR

This paper introduces a new algorithm for determining when two sets of finitely generated subgroups of a free group are related by an automorphism, and analyzes the structure of the subgroup fixing these sets using Outer space.

Contribution

It presents a novel algorithm based on Outer space for subgroup conjugacy problems in Out(F_n), and proves the subgroup fixing given subgroups is of type VF.

Findings

The subgroup fixing given subgroups is of type VF.

An algorithm to decide automorphism relations between subgroup sets.

A unified approach to automorphism and subgroup fixing problems.

Abstract

S. Gersten announced an algorithm that takes as input two finite sequences $\vec K=(K_1,\dots, K_N)$ and $\vec K'=(K_1',\dots, K_N')$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs: (1) $\mathsf{YES}$ or $\mathsf{NO}$ depending on whether or not there is an element $\theta\in \mathsf{Out}(F_n)$ such that $\theta(\vec K)=\vec K'$ together with one such $\theta$ if it exists and (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$. S. Kalajd\v{z}ievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler-Vogtmann's Outer space. New results include that the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$ is of type $\mathsf{VF}$, an equivariant version of these results, an application, and a unified approach to such questions.