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Quite free p-groups with trivial duality

Mohsen Asgharzadeh, Mohammad Golshani, Saharon Shelah

TL;DR

The paper advances the Trivial Duality Conjecture beyond torsion-free groups by developing a robust framework for p-groups that combines relative freeness with Shelah’s multi black box. Central to the approach are module-parameter tools, combinatorial bar-delta parameters, and generalized theta-fitness results, which yield relatively free abelian p-groups G with Hom(G, Z)=0 under suitable constraints. The authors define and utilize reduced and separable p-groups, culminating in a main theorem that constructs G with trivial duals while preserving high freeness, thereby extending Sh:1028-type results to a torsion setting. The work provides a path to resolving TDU for broader cardinalities and connects with classical p-group duality questions, offering concrete constructions and a generalized framework applicable to a wide class of abelian p-groups.

Abstract

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black box method and the application of specific homological properties of relative trees.

Quite free p-groups with trivial duality

TL;DR

The paper advances the Trivial Duality Conjecture beyond torsion-free groups by developing a robust framework for p-groups that combines relative freeness with Shelah’s multi black box. Central to the approach are module-parameter tools, combinatorial bar-delta parameters, and generalized theta-fitness results, which yield relatively free abelian p-groups G with Hom(G, Z)=0 under suitable constraints. The authors define and utilize reduced and separable p-groups, culminating in a main theorem that constructs G with trivial duals while preserving high freeness, thereby extending Sh:1028-type results to a torsion setting. The work provides a path to resolving TDU for broader cardinalities and connects with classical p-group duality questions, offering concrete constructions and a generalized framework applicable to a wide class of abelian p-groups.

Abstract

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black box method and the application of specific homological properties of relative trees.

Paper Structure

This paper contains 6 sections, 9 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Conveniences with freeness of trees and black box
  3. Freeness of relative trees
  4. The multi black box
  5. The relative notions of freeness and module parameters
  6. Trivial duality around reduced separable $p$-groups

Key Result

Theorem 1.5

Let $\mathbf x$ be a $\mathbf k$-combinatorial $\bar{\partial}$-parameter, $R=R_{\mathbf x}$ and let $(\chi, \bold d, \Xi)$ be a module problem. Suppose $\mathbf x$$\theta$-fits the triple $(\chi,\mathbf d,\Xi),\chi^+ \ge \theta + |R|^+$ and $\mathbf x$ has $\chi$-black box. The following assertions

Theorems & Definitions (37)

  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.8
  • Corollary 1.9
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Definition 2.8
  • Definition 2.9
  • ...and 27 more