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arXiv · 1509.06660

A complete solution of Markov's problem on connected group topologies

Dikran Dikranjan·Dmitri Shakhmatov
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Abstract

Every proper closed subgroup of a connected Hausdorff group must have index at least c, the cardinality of the continuum. 70 years ago Markov conjectured that a group G can be equipped with a connected Hausdorff group topology provided that every subgroup of G which is closed in all Hausdorff group topologies on G has index at least c. Counter-examples in the non-abelian case were provided 25 years ago by Pestov and Remus, yet the problem whether Markov's Conjecture holds for abelian groups G remained open. We resolve this problem in the positive.

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Dikran Dikranjan, Dmitri Shakhmatov. 2015-09-22. A complete solution of Markov's problem on connected group topologies. https://doi.org/10.1016/j.aim.2015.09.006

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