2023 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 / variety of groups / free group in a variety |
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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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| Causes of Carryover |
今年度も、コロナウイルスの感染関係で、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、海外出張を出来ず、使用額は286,000円に留まった。今年度は海外から共同研究者の招待を行う予定である。
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