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

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 th...

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Published in:arXiv.org 2022-10
Main Authors: Bestvina, Mladen, Feighn, Mark, Handel, Michael
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description 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. 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.
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title A McCool Whitehead type theorem for finitely generated subgroups of \(\mathsf{Out}(F_n)\)
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