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)，如果存在一个单位元的开邻域，它只包含一个平凡的子群。我们基于极小概周期性(MINAP)建立了两个性质的模型。(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G到C中包含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。我们称拓扑群G是MINAP模P当且仅当G的每个连续同态像都具有性质P。如果C是紧群类，则MINAP(C)是von Neumann的MINAP性质。类似地，如果P是平凡群的性质，那么MINAP模P与MinAP重合我们考虑了MINAP(Lie)，MINAP(NSS)和MINAP(LC)性质(LC代表局部紧)。MINAP(NSS)和MINAP(LC)都表示MINAP(LIE)，MINAP(LIE)表示MINAP属性。我们表明，所有这些影响都是不可逆转的。在Abel拓扑群中，Minap(LC)，Minap(Lie)和MINAP是等价的，而Abelian Minap(Nss)群的类与Dikranjan和Shakhmatov的SSGP(α)类的所有序数α的并重合.证明了如果P被连续同态保持，则群G是MINAP模P当且仅当G及其von Neumann核的商在其Bohr拓扑中具有性质P.给出了Abelian MINAP模P群的如下性质的代数结构：有限、有界、扭转、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.