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

一个拓扑群没有小子群（NSS），如果存在单位元的开邻域，除了平凡子群外不含其他子群。我们基于最小几乎周期性（MinAP）对两个属性进行建模。(A)设C是一类拓扑群。一个拓扑群G称为具有MinAP（C）性质，如果从G到包含在C中的群的任何非平凡同态是不连续的。(B)设P是拓扑群的一个性质。我们说一个拓扑群G是MinAP模P当且仅当G在一个紧群中的每个连续同态像都有性质P。如果C是紧群类，则MinAP（C）是冯·诺依曼的MinAP性质。类似地，如果P是平凡群的性质，则MinAP模P与MinAP一致我们考虑了MinAP（Lie），MinAP（NSS）和MinAP（LC）性质（LC代表局部紧）。MinAP（NSS）和MinAP（LC）都意味着MinAP（Lie），而MinAP（Lie）意味着MinAP属性。我们表明，这些影响都是可逆的。在Abel拓扑群中，MinAP（LC），MinAP（Lie）和MinAP是等价的，而Abel MinAP（NSS）群的类与Dikranjan和Shakhmatov的SSGP（α）类在所有序数α上的并是一致的.我们证明了：如果P由连续同态保持，则群G是MinAP模P当且仅当G与其von Neumann核的商在其Bohr拓扑中具有性质P.我们描述了Abel MinAP模P群的代数结构，其中P具有以下性质：有限，有界，挠，连通和紧。

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.