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A universal $P$-group of weight $\aleph$

Jan van Mill

TL;DR

This work identifies a CH-conditional universality phenomenon for $P$-groups: the $G_\delta$-modified homeomorphism group $\mathcal{H}(\omega^*)_{(\delta)}$ serves as a universal object for all $P$-groups of weight at most ${\mathfrak c}$, paralleling Parovičenko-type universality for spaces. The construction relies on embedding lemmas that pass from compact zero-dimensional $X$ to the Parovičenko remainder $\omega^*$ via a $eta$-extension, exploiting Rubin’s rigidity framework. The paper also clarifies the limitations: $\mathcal{H}(\omega^*)$ itself is not a $P$-group, and the universality result collapses in models where CH fails. Altogether, it positions $\mathcal{H}(\omega^*)$ as a structurally rich yet delicate universal object with deep ties to Parovičenko theory, non-archimedean groups, and the CH landscape. The results illuminate how set-theoretic assumptions shape universality phenomena for topological groups and highlight $\omega^*$-based constructions as a canonical testing ground for such universality questions.

Abstract

We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the Čech-Stone remainder of $ω$ with the $G_δ$-topology, is a universal object for all $P$-groups of weight at most ${\mathfrak c}$.

A universal $P$-group of weight $\aleph$

TL;DR

This work identifies a CH-conditional universality phenomenon for -groups: the -modified homeomorphism group serves as a universal object for all -groups of weight at most , paralleling Parovičenko-type universality for spaces. The construction relies on embedding lemmas that pass from compact zero-dimensional to the Parovičenko remainder via a -extension, exploiting Rubin’s rigidity framework. The paper also clarifies the limitations: itself is not a -group, and the universality result collapses in models where CH fails. Altogether, it positions as a structurally rich yet delicate universal object with deep ties to Parovičenko theory, non-archimedean groups, and the CH landscape. The results illuminate how set-theoretic assumptions shape universality phenomena for topological groups and highlight -based constructions as a canonical testing ground for such universality questions.

Abstract

We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the Čech-Stone remainder of with the -topology, is a universal object for all -groups of weight at most .
Paper Structure (12 sections, 15 theorems, 5 equations)

This paper contains 12 sections, 15 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topology
  4. Groups
  5. The topological groups $\mathcal{H}(X)$ for compact zero-dimensional $X$
  6. Teleman's Theorem and generalizations
  7. Rubin's Theorem
  8. Proofs of Theorems \ref{['eerstestellingzonder']} and \ref{['eerstecorzonder']}
  9. Proof of \ref{['tweedestelling']}
  10. Some topological properties of$\mathcal{H}(\omega^*)$
  11. Cardinal functions
  12. Universality properties of $\mathcal{H}(\omega^*)$.

Key Result

Theorem 1.1

Let $G$ be a topological group of weight ${\mathfrak c}$. Then there is a Parovičenko space $X$ such that $G_{(\delta)}$ is (topologically isomorphic to) a subgroup of $\mathcal{H}(X)$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2: $\mathsf{CH}$
  • Theorem 1.3
  • Theorem 2.1: Megrelishvili and Shlossberg MegrelishviliShlossberg12
  • Theorem 2.2: Rubin RubinM89
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 17 more