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

如果一个拓扑群存在一个开放邻域，除了平凡的子群外不包含其他子群，则该拓扑群不存在小子群（NSS）。我们基于最小几乎周期性（MinAP）建模了两个属性。(A)设C为一类拓扑群。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，则称拓扑群G具有MinAP(C)性质。(B)设P是拓扑群的一个性质。我们说拓扑群G是MinAP模P当且仅当G在紧群中的所有连续同态像都具有性质P。如果C是紧群的类，则MinAP(C)是von Neumann的MinAP性质。同样，如果P是平凡群的性质，则MinAP模P与minap重合。我们考虑了MinAP（Lie）、MinAP（NSS）和MinAP（LC）的性质（LC代表局部紧化）。MinAP（NSS）和MinAP（LC）都暗示MinAP(Lie), MinAP（Lie）暗示MinAP属性。我们表明，这些影响没有一个是可逆的。在Abelian拓扑群中，MinAP（LC）、MinAP（Lie）和MinAP是等价的，而Abelian MinAP（NSS）群的类别与Dikranjan和Shakhmatov的SSGP（α）类的所有序数α上的并集一致。证明了如果P被连续同态保存，则群G是MinAP模P当且仅当G的商及其von Neumann核在其玻尔拓扑中具有P的性质。我们描述了具有有限、有界、扭转、连通和紧性的Abelian MinAP模P群的代数结构。

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.