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2022 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 / Hausdorff embedding / extension of topologies
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.

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.

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.

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。

  • Research Products

    (2 results)

All 2023 2022

All Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (1 results) (of which Invited: 1 results)

  • [Journal Article] Star-covering properties and neighhbourhood assignments2023

    • Author(s)
      Fortunata Aurora Basile, Maddalena Bonanzinga, Fortunato Maesano, Dmitri B. Shakhmatov
    • Journal Title

      Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur.

      Volume: 101, no.1, A7 Pages: 1-14

    • DOI

      10.1478/AAPP.1011A7

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Zariski and (precompact) Markov topologies in free groups and their subgroups2022

    • Author(s)
      Dmitri Shakhmatov
    • Organizer
      Algebra, Topology and Their Interactions (Udine University, Italy)
    • Invited

URL: 

Published: 2023-12-25  

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