We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{\mathbb{Z}\left[t\right]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}\left[t\right]}$ containing $G$ is universally equivalent to $G$. Since finitely generated...

This paper investigates the productivity of the Zariski topology ${ℨ}_G$ of a group $G$. If $𝒢=\{G_i\mid i\in I\}$ is a family of groups, and $G={\prod }_{i\in I}{G}_i$ is their direct product, we prove that ${ℨ}_G\subseteq {\prod }_{i\in I}{ℨ}_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $𝒢=\{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., ${ℨ}_G={ℨ}_{G_1}\times {ℨ}_{G_2}$. 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 $G_1\times \left\{e_2\right\}$ is an elementary algebraic subset of $G_1\times G_2$ for every group $G_1$. We show among others, that $\Delta$ is stable...