2022 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 | Zariski topology / Markov topology / Hausdorff embedding / extension of topologies |
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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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| Causes of Carryover |
今年度も、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困難になっていため、海外出張を一回しか出来ず、使用額は304,980円になった。これからは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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