Abstract
A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points.
We prove that every product variety
U
V
\mathcal{UV}
, where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group.
In particular, there exist condensed groups of finite exponent.
As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in
U
V
\mathcal{UV}
.
A topological zero-one law and elementary equivalence of finitely generated groups
