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The construction principle and non homogeneity of uncountable relatively free groups

Davide Carolillo, Gianluca Paolini

TL;DR

This paper resolves Belegradek's question by showing that the construction principle $(\textbf{CP})$ yields non-$\aleph_1$-homogeneity for uncountable $V$-free groups, and extends beyond residual finiteness to varieties with torsion. The authors develop a framework based on infinitary logic, $V$-free bases, and the verbal functor to translate homogeneity questions into diagrammatic and elementary substructure arguments, producing a general criterion (Theorem) linking $(\textbf{CP})$ to non-homogeneity. They supply concrete methods to verify $(\textbf{CP})$ for broad varieties via Mekler-type results, and provide explicit coding constructions to exhibit pathological factorizations, thereby establishing non-$\aleph_1$-homogeneity for uncountable ranks and identifying elementary subgroups not occurring as $V$-free factors. The results significantly broaden the scope of non-homogeneity in relatively free groups and demonstrate the utility of combining infinitary logic with categorical and combinatorial tools in group variety theory.

Abstract

In [11] Sklinos proved that any uncountable free group is not $\aleph_1$-homogenenous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek's question in the negative. Our methods are general and they also applications in varieties with torsion, for example we show that if $V$ contains a non-solvable group, then any uncountable $V$-free group is not $\aleph_1$-homogenenous.

The construction principle and non homogeneity of uncountable relatively free groups

TL;DR

This paper resolves Belegradek's question by showing that the construction principle yields non--homogeneity for uncountable -free groups, and extends beyond residual finiteness to varieties with torsion. The authors develop a framework based on infinitary logic, -free bases, and the verbal functor to translate homogeneity questions into diagrammatic and elementary substructure arguments, producing a general criterion (Theorem) linking to non-homogeneity. They supply concrete methods to verify for broad varieties via Mekler-type results, and provide explicit coding constructions to exhibit pathological factorizations, thereby establishing non--homogeneity for uncountable ranks and identifying elementary subgroups not occurring as -free factors. The results significantly broaden the scope of non-homogeneity in relatively free groups and demonstrate the utility of combining infinitary logic with categorical and combinatorial tools in group variety theory.

Abstract

In [11] Sklinos proved that any uncountable free group is not -homogenenous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek's question in the negative. Our methods are general and they also applications in varieties with torsion, for example we show that if contains a non-solvable group, then any uncountable -free group is not -homogenenous.
Paper Structure (4 sections, 8 theorems, 15 equations)

This paper contains 4 sections, 8 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of the Main Theorem
  4. Applications

Key Result

Theorem 1.1

Let $V$ be a variety of groups. If $({\normalfont\textbf{CP}})$ holds in $V$, then for every uncountable $\kappa$, $F_V(\kappa)$ is not $\aleph_1$-homogeneous. Furthermore, $F_V(\aleph_0)$ has an elementary subgroup which is not a $V$-free factor of $F_V(\aleph_0)$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.8
  • proof
  • ...and 18 more