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Pré-Publication, Document De Travail Année : 2023

Centralisers and the virtually cyclic dimension of $\mathrm{Out}(F_N)$

Résumé

We prove that the virtually cyclic (geometric) dimension of the finite index congruence subgroup $\mathrm{IA}_N(3)$ of $\mathrm{Out}(F_N)$ is $2N-2$. From this we deduce the virtually cyclic dimension of $\mathrm{Out}(F_N)$ is finite. Along the way we prove Lück's property (C) holds for $\mathrm{Out}(F_N)$, we prove that the commensurator of a cyclic subgroup of $\mathrm{IA}_N(3)$ equals its centraliser, we give an $\mathrm{IA}_N(3)$ analogue of various exact sequences arising from reduction systems for mapping class groups, and give a near complete description of centralisers of infinite order elements in $\mathrm{IA}_3(3)$.
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Dates et versions

hal-04419042 , version 1 (26-01-2024)

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Yassine Guerch, Sam Hughes, Luis Jorge Sánchez Saldaña. Centralisers and the virtually cyclic dimension of $\mathrm{Out}(F_N)$. 2024. ⟨hal-04419042⟩
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