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Transactions of the American Mathematical Society

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

 

 

Abstract commensurators of profinite groups


Authors: Yiftach Barnea, Mikhail Ershov and Thomas Weigel
Journal: Trans. Amer. Math. Soc. 363 (2011), 5381-5417
MSC (2000): Primary 20E18; Secondary 22D05, 22D45
DOI: https://doi.org/10.1090/S0002-9947-2011-05295-5
Published electronically: March 28, 2011
MathSciNet review: 2813420
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Abstract: 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: y.barnea@rhul.ac.uk

Mikhail Ershov
Affiliation: Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, Virginia 22904-4137
Email: ershov@virginia.edu

Thomas Weigel
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi, 53, I-20125 Milan, Italy
Email: thomas.weigel@unimib.it

DOI: https://doi.org/10.1090/S0002-9947-2011-05295-5
Received by editor(s): September 1, 2009
Received by editor(s) in revised form: January 7, 2010
Published electronically: March 28, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.