2014 Fiscal Year Research-status Report
コンパクト型可換群の構造及びMarkov稠密性を実現する群位相の導入の研究
| Project/Area Number |
26400091
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| Research Institution | Ehime University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | connected space / Markov problem / unconditionally closed / abelian group / group topology / precompact group / minimal group |
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| Outline of Annual Research Achievements |
Every proper closed subgroup of a connected Hausdorff group must have index at least 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 continuum (in other words, all proper unconditionally closed subgroups of G have index at least continuum). 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. In a joint work with Dikranjan we resolved this problem in the positive; that is, we proved that an abelian group G admits a connected Hausdorff group topology if and only if for every proper unconditionally closed subgroup H of G the quotient group G/H has cardinality at least continuum. This result advances our understanding of the Markov-Zariski topology of an abelian group and its relation to the existence of connected Hausdorff group topologies.
Inspired by the above result, we have obtained complete characterizations of abelian groups that admit a precompact connected Hausdorff group topology and abelian groups that admit a minimal connected Hausdorff group topology. These results clarify the structure of compact-like connected groups.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
While working on main topics of this research project, we have discovered a new technique that, when combined with our own methods from previous publications, lead to a solution of seventy years old well-known problem of Markov on the existence of connected group topologies on abelian groups. Furthermore, the unique techniques we elaborated for the solution of Markov's problem lead us also to investigate (and subsequently completely solve) the Comfort-Protasov problem on the existence of minimally almost periodic group topologies on abelian groups. These unexpected new developments got us temporarily sidetracked from the original problems.
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| Strategy for Future Research Activity |
We shall finish the paper which provides a complete solution of the Comfort-Protasov problem on the existence of minimally almost periodic group topologies on abelian groups. We shall investigate the realization of the algebraic (Markov-Zariski) closure of a given countable subset of an abelian group in some pseudocompact group topology on this group.
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