Verbal functions of a group

Abstract

The aim of this paper is the study of elementary algebraic subsets of a group G, first defined by Markov in 1944 as the solution-set of a one-variable equation over G. We introduce the group of words over G, and the notion of verbal function of G in order to better describe the family of elementary algebraic subsets. The intersections of finite unions of elementary algebraic subsets are called algebraic subsets of G, and form the family of closed sets of the Zariski topology $\Zar_G$ on G. Considering only some elementary algebraic subsets, one can similarly introduce easier-to-deal-with topologies $\mathfrak T \subseteq \Zar_G$, that nicely approximate $\Zar_G$ and often coincide with it.