On the Absolutes of Compact Spaces with a Minimally Acting Group

If an w-bounded group G acts continuously on a compact Hausdorff space X and the orbit of every point is dense in X , then X iscoabsolute to a Cantor cube. A topological group G acts continuously on the topological space X if G is a group of homeomorphisms of X and the natural map X x G —> X is continuous. X is always assumed to be a compact Hausdorff space. We say that G acts minimally on X if the orbit {g(x): g £ G} is dense in X for every point x £ X. It is easy to see that G acts minimally on X if for every nonempty open subset U of X, there exist gx, ... , g„ £ G such that gx(U)U---Ug„(U) = X. Balear and Blaszczyk proved in [2] that every zero-dimensional compact space with a minimally acting countable group is coabsolute to a Cantor cube. Two compact spaces are said to be coabsolute if their respective Boolean algebras of regular open subsets are isomorphic. From results of Uspenskij [14] and Shapiro [12] it follows that if an («-bounded group G acts transitively and continuously on a compact space X, then X is coabsolute to a Cantor cube. A topological group G is said to be «-bounded if and only if for any neighbourhood U of its neutral element there is a countable subset A of G such that G = AU (Guran, see Archangelskij [1]). The aim of this note is to prove the following Theorem 1. If an co-bounded group G acts continuously and minimally on the compact Hausdorff space X, then X is coabsolute to a Cantor cube. The proof is based on results of Shapiro [12, 13] and makes use of arguments of Uspenskij [14] and Balear and Blaszczyk [2]. At first, we need the following outstanding result of Shapiro: (1) Every dyadic compact space which is homogeneous with respect to the weight is coabsolute to a Cantor cube. A space X is said to be homogeneous with respect to the weight if w(U) = w(X) for every nonempty open subset U. Received by the editors June 23, 1992. 1991 Mathematics Subject Classification. Primary 54D80, 22A05.