The algebraic structure of minimally almost periodic groups, extremely amenable groups and the Glasner-Pestov conjecture

最小几乎周期群、极其顺应群的代数结构和格拉斯纳-佩斯托夫猜想

• 批准号: 19J14198
• 负责人: YANEZ SALAZAR VICTOR H
• 金额: $ 1.34万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2019
• 资助国家: 日本
• 起止时间: 2019-04-25 至 2021-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-19J14198/
• 关键词: SSGP; MinAP; Lie group; Locally compact group; NSS group; Connected group; Bohr topology; algebraic SSGP; Divisible group; p-component; Rank of a group; Divisible rank

A topological group has no small subgroups (NSS) if there exists an open neighborhood of the identity containing no other subgroup but the trivial one. We modelled two properties based on minimal almost periodicity (MinAP).(A) Let C be a class of topological groups. A topological group G is said to have the MinAP(C) property if any non-trivial homomorphism from G to a group contained in C is discontinous. (B) Let P be a property of topological groups. We say that a topological group G is MinAP modulo P if and only if every continuous homomorphic image of G in a compact group has property P. If C is the class of compact groups, then MinAP(C) is the MinAP property of von Neumann. Similarly, if P is the property of being the trivial group, then MinAP modulo P coincides with MinAPWe considered the MinAP(Lie), MinAP(NSS) and MinAP(LC) properties (LC stands for locally compact). MinAP(NSS) and MinAP(LC) both imply MinAP(Lie), and MinAP(Lie) implies the MinAP property. We show that none of these implications are reversible. In Abelian topological groups, MinAP(LC), MinAP(Lie) and MinAP are equivalent, while the class of Abelian MinAP(NSS) groups coincides with the union over all ordinals α of the SSGP(α) classes of Dikranjan and Shakhmatov.We proved that if P is preserved by continuous homomorphisms, then a group G is MinAP modulo P if and only if the quotient of G with its von Neumann kernel has property P in its Bohr topology. We described the algebraic structure of Abelian MinAP modulo P groups for the following properties P: finite, bounded, torsion, connected and compact.

如果存在一个不包含其他亚组的身份的开放邻里，则拓扑组没有小的子组（NS）。我们基于最少的几乎周期性（MinAp）对两个属性进行了建模。（a）令C为一类拓扑组。据说拓扑组G具有MinAP（C）特性，如果从C中包含的G到一个组中的任何非平凡的同态性是不典型的。 （b）令P为拓扑组的属性。我们说，当且仅当紧凑型组中G的每个连续同构图像具有属性P时，拓扑组是Minap Modulo P。同样，如果p是琐碎群体的特性，则Minap Modulo P与Minapwe一致认为是Minap（Lie），MinAP（NSS）和MinAP（LC）特性（LC代表局部紧凑）。 Minap（NSS）和MinAP（LC）都表示Minap（Lie），Minap（Lie）表示Minap的性质。我们表明，这些含义都不可逆。 In Abelian topological groups, MinAP(LC), MinAP(Lie) and MinAP are equivalent, while the class of Abelian MinAP(NSS) groups coincides with the union over all ordinarys α of the SSGP(α) classes of Dikranjan and Shakhmatov.We proved that if P is Preserved by continuing homomorphisms, then a group G is MinAP modulo P if and only if the quotation G及其von Neumann内核具有其Bohr拓扑的财产P。我们描述了针对以下特性的Abelian Minap模量组的代数结构P：有限，有限，扭转，连接和紧凑。

项目成果

Subgroups of SSGP groups

SSGP 组的子组

• 发表时间: 2019
• 作者: Seiichiro Kondo;Kengo Hotate;Tosho Hirasawa;Masahiro Kaneko and Mamoru Komachi;Masahiro Kaneko and Danushka Bollegala;Yanez Salazar Victor Hugo;Masahiro Kaneko and Danushka Bollegala;Yanez Salazar Victor Hugo;高橋悠進，金子正弘，小町守;Yanez Salazar Victor Hugo;喜友名朝視顕，吉村綾馬，金子正弘，小町守;蘆田 真奈，平澤 寅庄，金子 正弘，小町 守;Yanez Salazar Victor Hugo;小山碧海，甫立健悟，金子正弘，小町守;Yanez Salazar Victor Hugo;今藤誠一郎，甫立健悟，平澤寅庄，金子正弘，小町守;Yanez Salazar Victor Hugo
• 通讯作者: Yanez Salazar Victor Hugo

Topological groups without infinite precompact continuous homomorphic images

没有无限预紧连续同态图像的拓扑群

• DOI: 10.1016/j.topol.2020.107544
• 发表时间: 2020
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: 吉村綾馬;金子正弘;梶原智之;小町守;Yanez Victor Hugo
• 通讯作者: Yanez Victor Hugo

SSGP topologies on free groups of infinite rank

无限秩自由群上的 SSGP 拓扑

• DOI: 10.1016/j.topol.2019.02.043
• 发表时间: 2019
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: Shakhmatov Dmitri;Yanez Victor Hugo
• 通讯作者: Yanez Victor Hugo

Direct sums of groups and ASSGP topologies

组和 ASSGP 拓扑的直和

• 发表时间: 2019
• 作者: Seiichiro Kondo;Kengo Hotate;Tosho Hirasawa;Masahiro Kaneko and Mamoru Komachi;Masahiro Kaneko and Danushka Bollegala;Yanez Salazar Victor Hugo;Masahiro Kaneko and Danushka Bollegala;Yanez Salazar Victor Hugo
• 通讯作者: Yanez Salazar Victor Hugo

Groups with many small subgroups, II

具有许多小子群的群，II

• 发表时间: 2019
• 期刊: Advances in General Topology and their Problems, RIMS Kokyuroku
• 作者: Shakhmatov Dmitri;Yanez Victor Hugo
• 通讯作者: Yanez Victor Hugo