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

在紧致性上具有不动点的拓扑群和群的潜在稠密子集

基本信息

  • 批准号:
    20K03615
  • 负责人:
  • 金额:
    $ 2.75万
  • 依托单位:
  • 依托单位国家:
    日本
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
  • 财政年份:
    2020
  • 资助国家:
    日本
  • 起止时间:
    2020-04-01 至 2025-03-31
  • 项目状态:
    未结题

项目摘要

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.
令G为一个组,然后用循环Z(其发电机用Z表示)在G的自由乘积G*z中的一个单词。方程w = 1的解集是g*z的所有元素x的集合,使得w'= 1,其中w'是通过用x在w中替换z的所有情况从w中获得的单词。 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如果H的Zariski(Markov)拓扑是它从G的Zariski(Markov)拓扑继承的Zariski(马尔可夫)。拓扑。我们证明,一个自由组的每个子组既是Zariski,又嵌入了Markov。另一方面,我们构建了一个自由组的正常子组,该子组带有2个发电机,而不是嵌入其中的Hausdorff。

项目成果

期刊论文数量(5)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Zariski and (precompact) Markov topologies in free groups and their subgroups
自由群及其子群中的 Zariski 和(预紧)马尔可夫拓扑
  • DOI:
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Shota Hoshinaga;Ikkei Hotta and Hiroshi Yanagihara;Dmitri Shakhmatov
  • 通讯作者:
    Dmitri Shakhmatov
Star-covering properties and neighhbourhood assignments
明星覆盖的房产和社区分配
Automorphism groups of dense subgroups of R^n
R^n 的稠密子群的自同构群
  • DOI:
    10.1016/j.topol.2019.107000
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    0.6
  • 作者:
    Vitalij A.Chatyrko;Dmitri B.Shakhmatov
  • 通讯作者:
    Dmitri B.Shakhmatov
On subsets of R^n spanning it via positive integers as multipliers
关于通过正整数作为乘数跨越 R^n 的子集
  • DOI:
    10.1016/j.topol.2020.107497
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0.6
  • 作者:
    Vitalij A.Chatyrko;Dmitri B.Shakhmatov
  • 通讯作者:
    Dmitri B.Shakhmatov
Unconditionally closed subsets of free (non-commutative) groups
自由(非交换)群的无条件闭子集
  • DOI:
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Ikkei Hotta;Sebastian Schleissinger and Toshiyuki Sugawa;Dmitri Shakhmatov
  • 通讯作者:
    Dmitri Shakhmatov
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D・B Shakhmatov其他文献

D・B Shakhmatov的其他文献

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具有动态拓扑结构和时延的多自主体系统的一致性控制
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