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arXiv · 2611.04581

Finiteness properties of centralisers in $\mathrm{Out}(F_N)$ and automorphisms of free-by-cyclic groups

Naomi Andrew·Yassine Guerch·Camille Horbez·Richard D. Wade
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Abstract

Let $F$ be a finitely generated free group. We prove that the centraliser of any element of $\mathrm{Out}(F)$ or $\mathrm{Aut}(F)$ is of type $\rm{VF}$. We also prove that the (outer) automorphism group of any free-by-cyclic group $M=F\rtimes_φ\mathbb{Z}$ is of type $\rm{VF}$, and so are the McCool subgroups of $\mathrm{Out}(M)$. This relies on the construction of a new $\mathrm{Out}(M)$-invariant splitting of $M$, that enables us to reduce the proof of the above statements to the case where the outer class of $φ$ has finite order.

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Naomi Andrew, Yassine Guerch, Camille Horbez, Richard D. Wade. 2026-11-03. Finiteness properties of centralisers in $\mathrm{Out}(F_N)$ and automorphisms of free-by-cyclic groups. https://arxiv.org/abs/2611.04581

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