Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isometry groups of Polish ultrametric spaces

Riccardo Camerlo, Alberto Marcone, Luca Motto Ros

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.

Isometry groups of Polish 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.

Paper Structure

This paper contains 19 sections, 31 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L$-trees
  4. Some useful functors
  5. The functor $\mathscr{U}$
  6. The functor $\mathscr{F}$
  7. The functor $\mathscr{G}$
  8. Generalized wreath products
  9. The classical definition
  10. Generalized wreath products over local domains
  11. Projective wreath products
  12. Some useful lemmas
  13. Main results
  14. Homogeneous spaces
  15. Exact spaces
  16. ...and 4 more sections

Key Result

Theorem 1.1

Up to (topological) isomorphism, the following classes of groups are the same: where the underlying order $\Delta$ of each wreath product is a countable $L$-tree, and $\mathbf{S}$ is a countable collection of local domains satisfying Holland's maximum condition.

Figures (2)

  • Figure 1: A graphical representation of an arbitrary $L$-tree, highlighting in particular the splitting level $\mathrm{split}(t,t')$ of the two $\leq_T$-incomparable nodes $t$ and $t'$.
  • Figure 2: The tree $\mathcal{T}_P$ is obtained by replacing each $z \in P$ with the tree depicted in the figure (with the order described in the text), where $(0,z)$ is identified with the original $z$. The nodes $z_\ell$ on the left branch are $\leq_T$-incomparable with every node which is not strictly $\leq_T$-above $t_z$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Example 3.4
  • Definition 3.5
  • Theorem 4.1
  • proof
  • Claim 4.1.1
  • ...and 76 more