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Isometry groups of proper metric spaces


Author: Piotr Niemiec
Journal: Trans. Amer. Math. Soc. 366 (2014), 2597-2623
MSC (2010): Primary 37B05, 54H15; Secondary 54D45
DOI: https://doi.org/10.1090/S0002-9947-2013-05941-7
Published electronically: October 28, 2013
MathSciNet review: 3165648
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Abstract: Given a locally compact Polish space $ X$, a necessary and sufficient condition for a group $ G$ of homeomorphisms of $ X$ to be the full isometry group of $ (X,d)$ for some proper metric $ d$ on $ X$ is given. It is shown that every locally compact Polish group $ G$ acts freely on $ G \times X$ as the full isometry group of $ G \times X$ with respect to a certain proper metric on $ G \times X$, where $ X$ is an arbitrary locally compact Polish space having more than one point such that $ (\mathrm {card}(G),\mathrm {card}(X)) \neq (1,2)$. Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space $ X$ having more than two points the set of all proper metrics $ d$ such that $ \mathrm {Iso}(X,d) = \{\mathrm {id}_X\}$ is dense in the space of all proper metrics on $ X$.


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Additional Information

Piotr Niemiec
Affiliation: Instytut Matematyki, Wydział Matematyki i Informatyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, 30-348 Kraków, Poland
Email: piotr.niemiec@uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-2013-05941-7
Keywords: Locally compact Polish group, isometry group, transitive group action, proper metric space, Heine-Borel metric space, proper metric, proper action.
Received by editor(s): January 26, 2012
Received by editor(s) in revised form: July 10, 2012
Published electronically: October 28, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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