Isometry groups of separable metric spaces

Abstract

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.

Generic elements in isometry groups of Polish ultrametric spaces

“…The next lemma is a simplified version of [5,Lemma 4.4]. Because y = f (y) for every y ∈ Y , x witnesses that there exists…”

“…It has been proved in [5] that for every Polish ultrametric space X there exists a complete pseudo-ultrametric d 1 on the set of orbits X/Iso(X) = { x : x ∈ X} defined by d 1 ( x , y ) = dist( x , y ).…”

“…Moreover, by [5,Lemma 4.4], every orbit x is closed, so X/Iso(X) = X/d 1 , where X/d 1 is the metric identification of d 1 . This gives rise to a projection π : X → X/d 1 defined by x → x .…”

“…for every x ∈ X; in particular, x α is closed for every x ∈ X; (4) X/d α is complete, and so it is a Polish ultrametric space; (5) Proof. Point (1) is clear, and Point (2) easily follows from the definition of pseudo-metric.…”

“…In the last section we analyze the canonical projection of a Polish ultrametric space X on the Polish ultrametric space of orbits of the natural action of Iso(X) on X, as described in [5]. Iterating this construction leads to a sequence of Polish ultrametric spaces, and their projections.…”