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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polish topometric groups


Authors: Itaï\ Ben Yaacov, Alexander Berenstein and Julien Melleray
Journal: Trans. Amer. Math. Soc. 365 (2013), 3877-3897
MSC (2010): Primary 03E15, 37B05
Published electronically: February 5, 2013
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Abstract: We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal's work as a guide, we explore the consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group $ \operatorname {Iso}(\mathbf {U}_1)$ of the bounded Urysohn space, the unitary group $ {\mathcal U}(\ell _2)$ of a separable Hilbert space, and the automorphism group $ \operatorname {Aut}([0,1],\lambda )$ of the Lebesgue measure algebra on $ [0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from $ \operatorname {Aut}([0,1],\lambda )$ into a separable topological group is continuous.


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Additional Information

Itaï\ Ben Yaacov
Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Alexander Berenstein
Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 # 18A-10, Bogotá, Colombia

Julien Melleray
Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05773-X
PII: S 0002-9947(2013)05773-X
Keywords: Ample generics, topometric group, automorphism group
Received by editor(s): July 8, 2011
Received by editor(s) in revised form: December 8, 2011
Published electronically: February 5, 2013
Additional Notes: Work on this project was facilitated by ANR chaire d’excellence junior THEMODMET (ANR-06-CEXC-007), a visit of the third author at the Erwin Schrödinger Institute in Vienna, and the ECOS Nord program (action ECOS Nord C10M01). The first author was also supported by the Institut Universitaire de France
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.