Isometry groups of Polish ultrametric spaces

TL;DR

This paper characterizes the isometry groups of Polish ultrametric spaces using generalized wreath products, solving longstanding problems and establishing connections with known group constructions in topology.

Contribution

It provides a complete characterization of isometry groups of Polish ultrametric spaces via generalized wreath products, addressing open problems from Krasner and Gao-Kechris.

Findings

Solved Krasner's open problem on isometry groups.

Established a correspondence with generalized wreath products.

Extended results to subclasses of ultrametric spaces.

Abstract

We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various kinds of generalized wreath products proposed in the literature by Hall, Holland, and Malicki.