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 $S_{∞}$ 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.
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We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.
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We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L ≤ Iso(X) and X is a locally compact separable metric space (in particular, for every abelian, quasi-countable group G), the following holds: G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.
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