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