Markov's problem for 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.
Keywords
Cite
@article{arxiv.2111.15277,
title = {Markov's problem for free groups},
author = {Dmitri Shakhmatov and Víctor Hugo Yañez},
journal= {arXiv preprint arXiv:2111.15277},
year = {2022}
}
Comments
The paper was re-organized into 7 sections (from original 11). A historical discussion in the Introduction has been enhanced. The statement of old Lemma 3.1 (new Lemma 3.3) was strengthened and its proof was simplified. Some questions from old Section 11 (new Section 7) were modified in order to incorporate known results in (new) Remark 7.3. Old question 11.7 has been solved in (new) Remark 4.4