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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Abstract commensurators of profinite groups
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by Yiftach Barnea, Mikhail Ershov and Thomas Weigel PDF
Trans. Amer. Math. Soc. 363 (2011), 5381-5417 Request permission


In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $\mathrm {Comm}(G)$ which depends only on the commensurability class of $G$. We study various properties of $\mathrm {Comm}(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $\mathrm {Comm}(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-$2$ completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, such as Pink’s analogue of Mostow’s strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups.
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Additional Information
  • Yiftach Barnea
  • Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
  • Email:
  • Mikhail Ershov
  • Affiliation: Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, Virginia 22904-4137
  • MR Author ID: 653972
  • Email:
  • Thomas Weigel
  • Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi, 53, I-20125 Milan, Italy
  • MR Author ID: 319262
  • Email:
  • Received by editor(s): September 1, 2009
  • Received by editor(s) in revised form: January 7, 2010
  • Published electronically: March 28, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 5381-5417
  • MSC (2000): Primary 20E18; Secondary 22D05, 22D45
  • DOI:
  • MathSciNet review: 2813420