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Transactions of the American Mathematical Society

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Commensurations of subgroups of $\mathrm {Out}(F_N)$


Authors: Camille Horbez and Richard D. Wade
Journal: Trans. Amer. Math. Soc. 373 (2020), 2699-2742
MSC (2010): Primary 20E08, 20E36, 20F28, 20F65
DOI: https://doi.org/10.1090/tran/7991
Published electronically: January 23, 2020
MathSciNet review: 4069231
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Abstract: A theorem of Farb and Handel [Publ. Math. Inst. Hautes Études Sci. 105 (2007), pp. 1–48] asserts that for $N\ge 4$, the natural inclusion from $\mathrm {Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where $N=3$. More generally, we give sufficient conditions on a subgroup $\Gamma$ of $\mathrm {Out}(F_N)$ ensuring that its abstract commensurator $\mathrm {Comm}(\Gamma )$ is isomorphic to its relative commensurator in $\mathrm {Out}(F_N)$. In particular, we prove that the abstract commensurator of the Torelli subgroup $\mathrm {IA}_N$ for all $N\ge 3$, or more generally any term of the Andreadakis–Johnson filtration if $N\ge 4$, is equal to $\mathrm {Out}(F_N)$. Likewise, if $\Gamma$ is the kernel of the natural map from $\mathrm {Out}(F_N)$ to the outer automorphism group of a free Burnside group of rank $N\geq 3$, then the natural map $\mathrm {Out}(F_N)\to \mathrm {Comm}(\Gamma )$ is an isomorphism.


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Additional Information

Camille Horbez
Affiliation: CNRS, Laboratoire de Mathématique d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, F-91405 Orsay, France
MR Author ID: 1008174
Email: camille.horbez@math.u-psud.fr

Richard D. Wade
Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
MR Author ID: 951412
Email: wade@maths.ox.ac.uk

Received by editor(s): February 22, 2019
Received by editor(s) in revised form: August 19, 2019, and August 22, 2019
Published electronically: January 23, 2020
Additional Notes: The first author acknowledges support from the Agence Nationale de la Recherche under Grant ANR-16-CE40-0006
Article copyright: © Copyright 2020 American Mathematical Society