Subgroups of monothetic groups

Embeddings into monothetic groups

We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption is essential.

Niemiec 的万有赋值阿贝尔群是 Lévy 群、强异类群、极可均群，且 $\mathbb{G}_r(0)$ 是泛单生成群

The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and $\mathbb{G}_r(0)$ is generically monothetic

本文证明 Niemiec 万有赋值阿贝尔群具有 Lévy、强异类、极可均等性质，并证明 $\mathbb{G}_r(0)$ 单生成，从而在 Urysohn 球面上构造出单生成群结构。

在单参数拓扑群中检测实数直线

Detecting the real line among one-parametric topological groups

本文研究了单参数拓扑群与实数直线的同构条件，证明了可度量且非单态的单参数拓扑群与实数直线同构，并探讨了博赫拓扑在非可度量设置下的应用。

局部紧严格凸度量群是交换群

Locally compact strictly convex metric groups are abelian

本文证明了局部紧严格凸度量群是交换群，通过构造实线到该群的同态嵌入并证明紧子群平凡性，结合经典结果完成证明。

Representing a profinite group as the homeomorphism group of a continuum

We contribute some information towards finding a general algorithm for constructing, for a given profinite group, $G$, a compact connected space, $X$, such that the full homeomorphism group, $H(X)$, with the compact-open topology is isomorphic to $G$ as a topological group. It is proposed that one should find a compact topological oriented graph $Γ$ such that $G\cong Aut(Γ)$. The replacement of the edges of $Γ$ by rigid continua should work as is exemplified in various instances where discrete graphs were used. It is shown here that the strategy can be implemented for profinite monothetic grou…

A note on the precompactness of weakly almost periodic groups

An action of a group $G$ on a compact space $X$ is called weakly almost periodic if the orbit of every continuous function on $X$ is weakly relatively compact in $C(X)$. We observe that for a topological group $G$ the following are equivalent: (i) every continuous action of $G$ on a compact space is weakly almost periodic; (ii) $G$ is precompact. For monothetic groups the result was previously obtained by Akin and Glasner, while for locally compact groups it has been known for a long time.