Семинар «Геометрия, топология и их приложения»

Let $G$ be a group. A subset of $X$ is said to be elementary algebraic, if it is the solution set on $G$ of a given equation of the form $g_{1} x^{\epsilon_{1}} g_{2} x^{\epsilon_{2}} \dots g_{n} x^{\epsilon_{n}} = 1$ for some $g_1, \dots , g_n \in G$ and integers $\epsilon_1, \dots , \epsilon_n \in \mathbb {Z}$. $X$ is algebraic whenever it is an intersection of a finite union of elementary algebraic subsets of $G$. The algebraic subsets of a group $G$ form a basis of closed sets for a unique topology on $G$ known as the Zariski topology of $G$. Meanwhile, the family of all subsets of $G$ which are closed in every Hausdorff group topology of $G$ form a family of closed subsets for another unique topology on $G$ known as the Markov topology of $G$. The Markov topology on a group is always finer than its Zariski topology.

A problem of Markov from 1945 asks whether each unconditionally closed subset of a group is always algebraic; equivalently whether the Markov and the Zariski topologies of a group must always coincide. In this talk we give an overview of current advances in the theory centered around Markov’s problem, and present a recent positive solution to Markov’s problem for the non-abelian free groups. The results presented during this talk were achieved jointly by Dmitri Shakhmatov (Ehime University, Japan) and the speaker.

Email address: victor yanez@comunidad.unam.mx