Non-locally compact Polish groups and two-sided translates of open sets

Maciej Malicki¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green St. Urbana, IL 61801, U.S.A. and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 200 (2008) no. 3, pp. 279-295

Résumé

This paper is devoted to the following question. Suppose that a Polish group $G$ has the property that some non-empty open subset $U$ is covered by finitely many two-sided translates of every other non-empty open subset of $G$. Is then $G$ necessarily locally compact? Polish groups which do not have the above property are called strongly non-locally compact. We characterize strongly non-locally compact Polish subgroups of $S_\infty $ in terms of group actions, and prove that certain natural classes of non-locally compact Polish groups are strongly non-locally compact. Next, we discuss applications of these results to the theory of left Haar null sets. Finally, we show that Polish groups such as the isometry group of the Urysohn space and the unitary group of the separable Hilbert space are strongly non-locally compact.

Détail

DOI : 10.4064/fm200-3-3

Keywords: paper devoted following question suppose polish group has property non empty subset covered finitely many two sided translates every other non empty subset necessarily locally compact polish groups which have above property called strongly non locally compact characterize strongly non locally compact polish subgroups infty terms group actions prove certain natural classes non locally compact polish groups strongly non locally compact discuss applications these results theory haar null sets finally polish groups isometry group urysohn space unitary group separable hilbert space strongly non locally compact