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}.

中文翻译：

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产品品种中的浓缩组

如果有限生成的标记群空间中的同构类没有孤立点，则称有限生成群 𝐺 是凝聚的。我们证明了每一个乘积变体 U⁢V\mathcal{UV}，其中𝒰（分别是𝒱）是一个非阿贝尔（分别是一个非局部有限）变体，包含一个凝聚群。特别是存在有限指数的凝聚群。作为应用，我们在 U⁢V\mathcal{UV} 中有限生成群的集合上获得了同构结构和初等等价关系的一些结果。