Productivity of the Zariski topology on groups

Dikranjan, D. ; Toller, D.

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 219-237

Résumé

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.

Détail

Classification : 20E22, 20F70, 20K25, 20K45, 22A05, 57M07
Keywords: Zariski topology; (elementary, additively) algebraic subset; $\delta$-word; universal word; verbal function; (semi) $\mathfrak Z$-productive pair of groups; direct product