The construction principle and non homogeneity of uncountable relatively free groups

In [11] Sklinos proved that any uncountable free group is not ${\mathrm{\aleph }}_1$-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 ${\mathrm{\aleph }}_1$-homogenenous.