2020 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 | automorphism group / general linear group / special orthogonal group / Euclid space / subsemigroup of R^n / generators |
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| Outline of Annual Research Achievements |
Let n be a positive integer. We study the automorphism group Aut(G) of a dense subgroup G of R^n. We show that Aut(G) can be naturally identified with the subgroupΦ(G) of the group GL(n,R) of all all non-degenerated square n-matrices A with real coefficients such that G A = G. We describe Φ(G) for many dense subgroups G of either the real line R or the plane R^2. We consider also an inverse problem of which symmetric subgroups of GL(n,R) can be realized as Φ(G) for some dense subgroup G of R^n.
For n>=2, we show that any proper subgroup H of GL(n,R) satisfying SO(n,R) ⊆ H cannot be realized in this way. (Here SO(n,R) denotes the special orthogonal group of dimension n.) We show that the realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.
For a positive integer n, we investigate subsets of R^n which generate it by the use of positive integers taken as multipliers, as well as a related question of which sub-semigroups of R^n are generated by different subsets of R^n. In particular, we characterize sigma-compact subsets of R^n generating it in this way, and show that this characterization does not hold for general subsets.
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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 is proceeding according to original plan.
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| Strategy for Future Research Activity |
We shall investigate the algebraic structure of topological abelian groups which have the algebraic small subgroup generating property. We hope to classify complely countable abelian groups which admit a group topology with the algebraic small subgroup generating property.
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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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