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

TL;DR

This paper proves Markov's conjecture for abelian groups, showing that such groups can be equipped with a connected Hausdorff topology if all their closed subgroups have large index, resolving a 70-year-old open problem.

Contribution

The paper provides a positive resolution of Markov's conjecture specifically for abelian groups, filling a long-standing gap in the understanding of connected group topologies.

Findings

Markov's conjecture holds for abelian groups

Connected Hausdorff topologies exist under the conjecture's conditions

The problem remains open for non-abelian groups

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.