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Transactions of the American Mathematical Society

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



On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

Authors: Antonios Manoussos and Polychronis Strantzalos
Journal: Trans. Amer. Math. Soc. 359 (2007), 5593-5609
MSC (2000): Primary 37B05, 54H20; Secondary 54H15
Published electronically: May 11, 2007
MathSciNet review: 2327044
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Abstract: We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action $ (G,X)$ we construct a zero-dimensional compactification $ \mu X$ of $ X$ with the properties: (a) there exists an extension of the action on $ \mu X$, (b) if $ \mu L\subseteq \mu X\setminus X$ is the set of the limit points of the orbits of the initial action in $ \mu X$, then the restricted action $ (G,\mu X\setminus \mu L)$ remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on $ \mu X\setminus \mu L$ by that of $ \mu X$, and (c) $ \mu X$ is the maximal among the zero-dimensional compactifications of $ X$ with these properties. Proper actions are usually embedded in the endpoint compactification $ \varepsilon X$ of $ X$, in order to obtain topological invariants concerning the cardinality of the space of the ends of $ X$, provided that $ X$ has an additional ``nice" property of rather local character (``property Z", i.e., every compact subset of $ X$ is contained in a compact and connected one). If the considered space has this property, our new compactification coincides with the endpoint one. On the other hand, we give an example of a space not having the ``property Z" for which our compactification is different from the endpoint compactification. As an application, we show that the invariant concerning the cardinality of the ends of $ X$ holds also for a class of actions strictly containing the properly discontinuous ones and for spaces not necessarily having ``property Z".

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Additional Information

Antonios Manoussos
Affiliation: Fakultät für Mathematik, SFB 701, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Polychronis Strantzalos
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis, GR-157 84, Athens, Greece

Keywords: Proper actions, properly discontinuous actions, equicontinuous actions, indivisibility, zero-dimensional compactifications, inverse systems.
Received by editor(s): December 9, 2005
Published electronically: May 11, 2007
Additional Notes: This work was partially supported by DFG Forschergruppe “Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik" and SFB 701 "Spektrale Strukturen und Topologische Methoden in der Mathematik" at the University of Bielefeld, Germany.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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