2021 Fiscal Year Research-status Report
Topological groups with fixed point on compacta property and potentially dense subsets of groups
| Project/Area Number |
20K03615
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| Research Institution | Ehime University |
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Principal Investigator |
D・B Shakhmatov 愛媛大学, 理工学研究科(理学系), 教授 (90253294)
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| Project Period (FY) |
2020-04-01 – 2025-03-31
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| Keywords | free group / algebraic set / Zariski topology / unconditionally closed / Markov topology / precompact topology |
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| Outline of Annual Research Achievements |
A subset of a group G is unconditionally closed in G if it is closed in every Hausdorff group topology on G. The family of all unconditionally closed subsets of G forms the family of closed subsets of a unique topology on G called its Markov topology. Similarly, a family of subsets of G which are closed in each precompact group topology on G coincides with the family of closed subsets of the so-called precompact Markov topology of G. We prove that every unconditionally closed subset of a free group is algebraic, thereby answering a problem of Markov for free groups. In modern terminology this means that Markov and Zariski topologies coincide for free groups. Moreover, we show that for non-commutative free groups, Markov topology differs from precompact Markov topology. This is accomplished by finding a sequence S in the free group F with two generators which converges to the identity in each precompact Hausdorff group topology on F (and thus, in the precompact Markov topology on F), yet there exists a Hausdorff group topology on F such that S does not converge to the identity with respect to this topology (and thus, S is not closed in the Markov topology of G). We also deduce from our results that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The research proceeds according to the original plan.
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| Strategy for Future Research Activity |
We shall investigate whether a subgroup of a free group is always Zariski embedded, Markov embedded and/or Hausdorff embedded into the whole group. We shall also attempt to characterize potentially dense subsets of countable free groups.
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| Causes of Carryover |
コロナウイルスの感染拡大によって、海外出張や海外から研究者の招待は不可能になったため、使用額は0になった。感染状況は改善すれば海外出張や海外から研究者の招待を行う予定である。
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