Condensed groups in product varieties

TL;DR

This paper proves that certain product varieties of groups always contain condensed groups, which are groups with a highly non-isolated isomorphism class, and explores implications for the structure of finitely generated groups.

Contribution

It establishes the existence of condensed groups within specific product varieties of groups, including those of finite exponent, and analyzes related structural properties.

Findings

Existence of condensed groups in product varieties where one factor is non-abelian and the other non-locally-finite.

Condensed groups of finite exponent exist within these varieties.

Results on the isomorphism relation and elementary equivalence among finitely generated groups in these 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 $\mathcal{UV}$, 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 $\mathcal{UV}$.