The construction principle and non homogeneity of uncountable relatively free groups

TL;DR

This paper demonstrates that uncountable relatively free groups, including those with torsion, are not $eth_1$-homogeneous, extending previous results and using methods from infinitary logic.

Contribution

It shows that the non-homogeneity property holds for all uncountable relatively free groups, removing the residual finiteness assumption from prior work.

Findings

Uncountable relatively free groups are not $eth_1$-homogeneous.

The non-homogeneity result applies to groups with torsion.

Methods from infinitary logic are used to establish these results.

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.