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

Abstract

S. Gersten announced an algorithm that takes as input two finite sequences $\vec{K}=(K_1,\dots ,K_N)$ and ${\vec{K}}^{\prime }=(K_1^{\prime },\dots ,K_N^{\prime })$ 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 $\theta (\vec{K})={\vec{K}}^{\prime }$ 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. Kalajdžievski 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.

33 pages, 3 figures. Revised taking into account referee's comments. To appear in AHL

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