Topological transformation groups

A topological transformation group without non-trivial equivariant compactifications

“…This is a result by Stojanov [22]. This motivated Megrelishvili to ask whether the same conclusion holds for the unit sphere in every separable reflexive Banach space E under the action of the group of isometries of E with the strong operator topology, see Question 2.5 (2005) in [14]. However, it appears to us the question needs to be adjusted.…”

“…6.8. In conclusion, here is a question suggested by Furstenberg and Scarr (see [14], question 2.6 (2006)): does there exist a topological transformation group (G, X) whose universal equivariant compactification is a singleton, yet the action of G on X is transitive?…”

“…For a more detailed treatment with numerous related results, references, and open problems, see Megrelishvili's survey paper [14].…”

“…The question was apparently never mentioned in Smirnov's papers; however, it was well known among the Moscow general and geometric topologists, and later documented in Megrelishvili's papers [13,14]. (See also a discussion in [19], Rem.…”

“…An equivariant compactification of X is a compact space K, equipped with a continuous action by G, together with a continuous equivariant map i : X → K having a dense image in K. (See [23,24,25,22,12,14,16].) The question of interest is whether, given G and X, there exists a compactification into which X embeds topologically, that is, i is a homeomorphism onto its image.…”