Abstract. We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group. IntroductionFor a metric space (X, d), let Iso(X) be the group of all isometries of X equipped with the pointwise convergence topology.The first part of the paper is concerned with representing groups as full isometry groups of metric spaces so that nice properties of the group are reflected by nice properties of the metric space. It is easy to see that if X is a Polish metric space, then Iso(X) is Polish. Again, it is an easy observation that Iso(X) is compact provided that X is compact. It was proved in [4] that Iso(X) is locally compact if X is proper, that is, if all closed balls of (X, d) are compact. A natural question arises whether the converses to these facts hold.In [4], Gao and Kechris showed that every Polish group is indeed isomorphic to the isometry group of some Polish space. Then Melleray [7] found a simpler proof of their result and used it to prove that every compact group is isomorphic to the isometry group of a compact space. In Section 2 Theorem 2.1, we provide the last missing piece of the picture by showing that every locally compact Polish group is the isometry group of a proper Polish space. This solves a problem posed by Gao and Kechris in [4, p.76].In Section 3, we comment on the tools used in the proof of the main result from the previous section.2000 Mathematics Subject Classification. 22D12, 54E35, 54H15.

Abstract. We study abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with an invariant metric or as quasicountable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product topology.We show, among other results, 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 (e.g., G is abelian, quasi-countable), 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.

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

We prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.