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

Bestvina, Mladen; Feighn, Mark; Handel, Michael

doi:10.5802/ahl.159

Bestvina, Mladen; Feighn, Mark; Handel, Michael

A McCool Whitehead type theorem for finitely generated subgroups of

Out (F_{n})

Annales Henri Lebesgue, Volume 6 (2023), pp. 65-94.

Metadata

DOI10.5802/ahl.159

Keywords free group, geometric group theory

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) YES or NO depending on whether or not there is an element $θ \in Out (F_{n})$ such that $θ (\vec{K}) = {\vec{K}}^{'}$ together with one such $θ$ if it exists and
(2) a finite presentation for the subgroup of $Out (F_{n})$ 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 $Out (F_{n})$ fixing $\vec{K}$ is of type $VF$ , an equivariant version of these results, an application, and a unified approach to such questions.

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