D. Dikranjan, D. Toller
Productivity of the Zariski topology on groups

Comment.Math.Univ.Carolin. 54,2 (2013) 219-237.

Abstract:This paper investigates the productivity of the Zariski topology $\mathfrak Z_G$ of a group $G$. If $\mathcal G = \{G_i\mid i\in I\}$ is a family of groups, and $G = \prod_{i\in I}G_i$ is their direct product, we prove that $\mathfrak Z_G\subseteq \prod_{i\in I}\mathfrak Z_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $\mathcal G = \{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., $\mathfrak Z_G = \mathfrak Z_{G_1}\times \mathfrak Z_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta$ of groups $G_2$ such that $G_1\times \{e_{2}\}$ is an elementary algebraic subset of ${G_1\times G_2}$ for every group $G_1$. We show among others, that $\Delta$ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to $\Delta$. In particular, $\Delta$ contains arbitrary direct products of free non-abelian groups.

Keywords: Zariski topology, (elementary, additively) algebraic subset, $\delta$-word, universal word, verbal function, (semi) $\mathfrak Z$-productive pair of groups, direct product
AMS Subject Classification: 20F70 20K45 20K25 57M07