Markov's problem for free groups

Dmitri Shakhmatov; Víctor Hugo Yañez

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arXiv · 2111.15277

Markov's problem for free groups

Dmitri Shakhmatov·Víctor Hugo Yañez

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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.

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Dmitri Shakhmatov, Víctor Hugo Yañez. 2022-10-17. Markov's problem for free groups. https://arxiv.org/abs/2111.15277

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