A topological transformation group without non-trivial equivariant compactifications

Abstract

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.

Maximal equivariant compactifications

“…Pestov raised several questions in [38] about a possible coincidence between the maximal G-compactification and the Gromov compacfification for some natural geometrically defined isometric actions (Urysohn sphere and Gurarij sphere, among others). These problems were studied recently in [16] (with a positive answer in the case of the Urysohn sphere and a negative answer for the Gurarij sphere).…”

“…Question 6.3. (H. Furstenberg and T. Scarr (see [31,38])) Let G×X → X be a continuous action with one orbit (that is, algebraically transitive) of a (metrizable) topological group G on a (metrizable) space X. Is it true that X is G-Tychonoff ?…”

“…See, for example, R. Brook [8], J. de Vries [50,51,52,53,54,55], Yu.M. Smirnov [41,42,43,44], Antonyan-Smirnov [4], Smirnov-Stoyanov [45], L. Stoyanov [47,48], M. Megrelishvili [24,25,26,27,28,30,31,32], Dikranjan-Prodanov-Stoyanov [10], Megrelishvili-Scarr [34], V. Uspenskij [49], S. Antonyan [2], Gonzalez-Sanchis [14], V. Pestov [37,38], J. van Mill [35], A. Sokolovskaya [46], Google-Megrelishvili [15], Kozlov-Chatyrko [22], N. Antonyan, S. Antonyan and M. Sanchis [3], K. Kozlov [18,19,20,21], N. Antonyan [1], Karasev-Kozlov [17], Ibarlucia-Megrelishvili [16] (and many additional references in these publications).…”

“…Moreover, answering an old problem due to Smirnov, an extreme example was found by V. Pestov [38] by constructing a countable metrizable group G and a countable metrizable non-trivial G-space X for which every equivariant compactification is a singleton.…”

Maximal equivariant compactification of the Urysohn spaces and other metric structures

“…However, this fails in general, as first shown by the second author [24] (resolving a question of de Vries [41]), who built a Polish fan X together with a Polish group G ≤ Homeo(X) such that the system G X has no injective G-compactifications. Recently, and answering an old question of Smirnov, Pestov [31] exhibited an extreme counterexample by constructing a countable metrizable group G and a countable metrizable non-trivial G-space X for which every equivariant compactification is trivial, i.e., a singleton. The example is obtained by a clever iteration of the construction of [24].…”