| 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 2023)
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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 / variety of groups / free group in a variety / 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 |
A variety of groups is a class of groups which is closed with respect to subgroups, direct products and homomorphic images. Given a variety V of groups and a subset X of a group G, we say that G is V-free over X if G belongs to V, and every map f from X to a group H from the variety V admits a unique extension to a homomorphism from G to H. A group G is V-free if it is V-free over some of its subsets.
We prove that Markov and Zariski topologies coincide for V-free groups, for every variety V of groups, thereby solving 79 years old problem of Markov for V-free groups. When V is the variety of all groups, this implies that all free groups have coinciding Markov and Zariski topologies. This particular case was obtained earlier by the author and Victor Hugo Yanez. The key to the proof of main result is the following theorem. For every countable subset Y of a set X, every Hausdorff group topology on the V-free group with alphabet Y can be extended to a Hausdorff group topology on the V-free group with alphabet X. (Here V is an arbitrary variety of groups.)
We expect that new technique developed for proving these results would help to find a characterization of countable Zariski dense sets in free (and more generally, V-free) groups.
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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 original plan.
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| Strategy for Future Research Activity |
We shall attempt to characterize potentially dense subsets of countable free groups, as well as V-free groups in a given variety V.
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