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$\aleph_1$-free abelian non-Archimedean Polish groups

Gianluca Paolini, Saharon Shelah

TL;DR

The paper investigates uncountable $\aleph_1$-free abelian non-Archimedean Polish groups, establishing the existence of continuum many separable examples that are not topologically product groups and are pairwise non-image, non-isomorphic, via an engine that builds inverse limits of completely decomposable TFAB components. It also shows that the four properties $\mathbb{Z}$-homogeneity, $\aleph_1$-freeness, separability and torsionlessness are complete co-analytic on the space of closed abelian subgroups of $S_\infty$, using descriptive-set-theoretic reductions from trees to closed subgroups. The engine and the notion of nice pairs provide fine control over torsion properties and separability, yielding both existence and separation results, as well as counterexamples illustrating the boundaries of these notions. Additionally, the paper offers independent constructions separating $\mathbb{Z}$-homogeneous and $\aleph_1$-free, and demonstrates limitations of candidate sufficient conditions for $\aleph_1$-freeness via tree-based counterexamples. This work deepens the structural understanding of non-Archimedean Polish groups from both algebraic and descriptive-set-theoretic perspectives, with potential implications for classification and representation problems in this domain.

Abstract

An uncountable $\aleph_1$-free group can not admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed e.g. by the Baer-Specker group $\mathbb{Z}^ω$, in fact, more strongly, $\mathbb{Z}^ω$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are not topologically isomorphic to product groups and are pairwise not continuous homomorphic images of each other. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freeness and $\mathbb{Z}$-homogeneity.

$\aleph_1$-free abelian non-Archimedean Polish groups

TL;DR

The paper investigates uncountable -free abelian non-Archimedean Polish groups, establishing the existence of continuum many separable examples that are not topologically product groups and are pairwise non-image, non-isomorphic, via an engine that builds inverse limits of completely decomposable TFAB components. It also shows that the four properties -homogeneity, -freeness, separability and torsionlessness are complete co-analytic on the space of closed abelian subgroups of , using descriptive-set-theoretic reductions from trees to closed subgroups. The engine and the notion of nice pairs provide fine control over torsion properties and separability, yielding both existence and separation results, as well as counterexamples illustrating the boundaries of these notions. Additionally, the paper offers independent constructions separating -homogeneous and -free, and demonstrates limitations of candidate sufficient conditions for -freeness via tree-based counterexamples. This work deepens the structural understanding of non-Archimedean Polish groups from both algebraic and descriptive-set-theoretic perspectives, with potential implications for classification and representation problems in this domain.

Abstract

An uncountable -free group can not admit a Polish group topology but an uncountable -free abelian group can, as witnessed e.g. by the Baer-Specker group , in fact, more strongly, is separable. In this paper we investigate -free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so -free) abelian non-Archimedean Polish groups which are not topologically isomorphic to product groups and are pairwise not continuous homomorphic images of each other. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of : separability, torsionlessness, -freeness and -homogeneity.
Paper Structure (11 sections, 7 theorems, 11 equations)

This paper contains 11 sections, 7 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. The descriptive set theory setting
  3. A few crucial claims on $\mathrm{TFAB}$'s
  4. The engine
  5. Nice pairs
  6. Proofs of main theorems
  7. Proof of Theorem \ref{['second_th']}
  8. Proof of Theorem \ref{['being_almost_free_coanalytic']}
  9. Other results
  10. Separating $\mathbb{Z}$-homogeneous and $\aleph_1$-free
  11. For every $\eta \in [T]$, $\bigcap \{L(\eta \restriction n)\}$ is finite is not sufficient

Key Result

Theorem 1.2

There are continuum many separable (hence torsionless, hence $\aleph_1$-free) abelian non-Archimedean Polish groups which are not topologically isomorphic to product groups and are pairwise not continuous homomorphic images of each other. Furthermore, all these groups can be taken to be inverse limi

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.4
  • Definition 2.10
  • Proposition 2.12
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 31 more