Topological groups with fixed point on compacta property and potentially dense subsets of groups
| Project/Area Number |
20K03615
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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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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| Project Status |
Granted (Fiscal Year 2022)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2024: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | Zariski topology / Markov topology / Hausdorff embedding / extension of topologies / free group / algebraic set / unconditionally closed / precompact topology / automorphism group / general linear group / special orthogonal group / Euclid space / subsemigroup of R^n / generators / 位相群 / 極小概周期群 / extreme amenability / 稠密可能な集合 / 無条件閉集合 |
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| Outline of Research at the Start |
コンパクト群への非自明な連続準同型写像をもたない位相群を極小概周期群とよぶ. 任意のコンパクト空間への連続群作用が不動点をもつ位相群を超従順群という. 超従順群は極小概周期群であるが, 可換な極小概周期群が超従順群であるか否かは未解決である. 特に, 可分位相群に対するこの問題は, 整数論と密接な関係をもつ. 一方, 1941年のMarkovによる未解決問題「群Gのザリスキー位相で稠密な部分集合はG上のあるハウスドルフ群位相で稠密であるか」はよく知られている. 本研究では, 可換捩れ群に対するMarkovの問題の解決と, 可換群における極小概周期群と超従順群の関係の解明を目指す.
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| Outline of Annual Research Achievements |
Let G be a group, and let w be a word in the free product G*Z of G with the cyclic group Z (whose generator is denoted by z). The solution set of an equation w=1 is the set of all elements x of G*Z such that w'=1, where w' is the word obtained from w by replacing all occurencies of z in w with x. The Zariski (verbal) topology of a group G is the smallest topology on G in which solution set of all equations w=1 in G are closed.
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.
A subgroup H of a group G is Zariski (Markov) embedded in G if the Zariski (Markov) topology of H is the subspace topology it inherits from the Zariski (Markov) topology of G. A subgroup H of a group G is Hausdorff embedded in G if every Hausdorff group topology on H can be extended to a Hausdorff group topology of G in such a way that the original topology becomes a subgroup topology. We prove that every subgroup of a free group is both Zariski and Markov embedded in it. On the other hand, we construct a normal subgroup of a free group with 2 generators which is not Hausdorff embedded in it.
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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 attempt to characterize potentially dense subsets of countable free groups. In a given variety V of groups, we shall attempt to prove that the free group in the variety V has its Markov and Zariski topologies coincide.
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Report
(3 results)
Research Products
(5 results)
