arXiv:2111.15277 Abstract | arXiv Analytics

arXiv:2111.15277 [math.GR]Abstract References Reviews Resources

Markov's problem for free groups

Published 2021-11-30, updated 2022-01-19Version 2

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.

Comments: Sections 2 and 11 have been added

Categories: math.GR, math.AG, math.CO, math.GN, math.LO

Subjects: 20E05, 03C55, 03E05, 03E75, 14L99, 20E06, 22A05, 54E50, 54H11

Keywords: markovs problem, zariski topologies coincide, precompact markov topology, zariski topologies differ, old problem

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