A topological transformation group without non-trivial equivariant compactifications

TL;DR

This paper constructs a countable metrizable group acting on the rationals such that the only equivariant compactification is trivial, answering a question posed in the 1980s about the existence of such groups.

Contribution

It provides the first example of a topological transformation group with no non-trivial equivariant compactifications, using a recursive construction based on Megrelishvili's metric fan.

Findings

Constructed a countable metrizable group with trivial equivariant compactification.

Answered a long-standing question by Smirnov from the 1980s.

Demonstrated a method to produce such groups using recursive applications of known constructions.

Abstract

There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to Megrelishvili, which is a metric fan equipped with a certain group of homeomorphisms. The question of existence of a topological transformation group with the property in the title was asked by Yu.M. Smirnov in the 1980s.