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AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

Part of: Connections with other structures, applications Set theory Topological dynamics

Published online by Cambridge University Press:  11 April 2017

Marcin Sabok
Marcin Sabok*
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, H3A 0B9Canada Instytut Matematyczny Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-956 Warszawa, Poland (marcin.sabok@mcgill.ca)
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Abstract

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Keywords

automatic continuityPolish groupsisometry groupsUrysohn space

MSC classification

Primary: 03E15: Descriptive set theory 54H11: Topological groups 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 3 , May 2019 , pp. 561 - 590
Copyright
© Cambridge University Press 2017 

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Footnotes

The author acknowledges partial support from the following sources: the NCN (the Polish National Science Centre) grant 2012/05/D/ST1/03206, the MNiSW (the Polish Ministry of Science and Higher Education) grant 0435/IP3/2013/72 and the ‘Mobilność Plus’ program of the MNiSW through grant 630/MOB/2011/0.

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