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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Commensurations of subgroups of $\mathrm {Out}(F_N)$
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by Camille Horbez and Richard D. Wade PDF
Trans. Amer. Math. Soc. 373 (2020), 2699-2742 Request permission


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:
  • Richard D. Wade
  • Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
  • MR Author ID: 951412
  • Email:
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 2699-2742
  • MSC (2010): Primary 20E08, 20E36, 20F28, 20F65
  • DOI:
  • MathSciNet review: 4069231