Markov's problem for free groups

TL;DR

This paper proves that in free groups, unconditionally closed sets are algebraic, confirming Markov's long-standing conjecture and showing that Markov and Zariski topologies are equivalent in this context.

Contribution

It resolves a 76-year-old problem by establishing the equivalence of Markov and Zariski topologies in free groups, and explores their differences with the precompact Markov topology.

Findings

Markov and Zariski topologies coincide in free groups

Unconditionally closed sets are algebraic in free groups

Markov and Zariski topologies differ from the precompact Markov topology in non-commutative free groups

Abstract

We prove that every unconditionally closed subset of a free group is algebraic, thereby answering affirmatively a 76 years old problem of Markov for free groups. In modern terminology, this means that Markov and Zariski topologies coincide in free groups. It follows that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients. We also show that Markov and Zariski topologies differ from the so-called precompact Markov topology in non-commutative free groups.