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Isometry groups of separable metric spaces

Published online by Cambridge University Press:  01 January 2009

MACIEJ MALICKI  and
SŁAWOMIR SOLECKI
MACIEJ MALICKI
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA. e-mail: mamalicki@gmail.com; ssolecki@math.uiuc.edu
SŁAWOMIR SOLECKI
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA. e-mail: mamalicki@gmail.com; ssolecki@math.uiuc.edu
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Abstract

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 67 - 81
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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