Compactifications of topological groups

TL;DR

This paper explores various natural compactifications of topological groups, such as the greatest ambit, Roelcke, and weakly almost periodic compactifications, and discusses their properties and implications for group dynamics and minimality.

Contribution

It provides a detailed analysis of different compactifications of topological groups and their applications to properties like extreme amenability and minimality.

Findings

Roelcke compactification can be a singleton for certain groups

The greatest ambit relates to right uniformly continuous functions

Roelcke compactification helps prove minimality of groups

Abstract

Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on $G$; (2) the Roelcke compactification $R(G)$ corresponds to the algebra of functions which are both left and right uniformly continuous; (3) the weakly almost periodic compactification $W(G)$ is the envelopping compact semitopological semigroup of $G$ (`semitopological' means that the multiplication is separately continuous). The universal minimal compact $G$-space $X=M_G$ is characterized by the following properties: (1) $X$ has no proper closed $G$-invariant subsets; (2) for every compact $G$-space $Y$ there exists a $G$-map $X\to Y$. A group $G$ is extremely amenable, or has the fixed point on compacta property, if $M_G$ is a singleton. We discuss some results and questions by V. Pestov and E. Glasner on extremely amenable groups. The Roelcke compactifications were used by M. Megrelishvili to prove that $W(G)$ can be a singleton. They can be used to prove that certain groups are minimal. A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology.