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2023 Fiscal Year Research-status Report

Topological groups with fixed point on compacta property and potentially dense subsets of groups

Research Project

Project/Area Number 20K03615
Research InstitutionEhime University

Principal Investigator

D・B Shakhmatov  愛媛大学, 理工学研究科(理学系), 教授 (90253294)

Project Period (FY) 2020-04-01 – 2025-03-31
KeywordsZariski topology / Markov topology / variety of groups / free group in a variety
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.

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.

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.

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。

  • Research Products

    (1 results)

All 2024

All Presentation (1 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results)

  • [Presentation] Markov and Zariski topologies in free groups in a given variety2024

    • Author(s)
      Dmitri Shakhmatov
    • Organizer
      Ning-jin-tai Topology Workshop (in honor of Professor Wei He’s 60th anniversary) , Taizhou University, Taizhou (Jiangsu Province), China
    • Int'l Joint Research / Invited

URL: 

Published: 2024-12-25  

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