Condensed groups in product varieties

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 $\mathcal{U}\mathcal{V}$, where $\mathcal{U}$ (respectively, $\mathcal{V}$) 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 relation and elementary equivalence on the set of finitely generated groups in $\mathcal{U}\mathcal{V}$.

Some typos are fixed. To appear in the Journal of Group Theory

References in corpus (1)

• A topological zero-one law and elementary equivalence of finitely generated groups