Constructing Tychonoff G-spaces which are not G-Tychonoff

Abstract

Jan de Vries' compactification problem is whether every Tychonoff G-space can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V-group. De Vries showed that every locally compact group G is a V-group. The first example of a non- V-group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing noncompactifiable G-spaces. We show that the class of non- V-groups is large and contains all second countable (even ℵ 0- bounded ) nonlocally precompact groups. This establishes the existence of monothetic (even cyclic) non- V-groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of “ G-normality”.