The Group of Isometries of a Locally Compact Metric Space with One End

In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X, d) with one end. Using the notion of pseudocomponents introduced by S. Gao and A. S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not compact andG acts properly on this pseudo-component. The complement of the non-compact component is a compact subset of X and G may fail to act properly on it. 1. Preliminaries and the main result The idea to study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X, d) with one end, using the notion of pseudo-components introduced by S. Gao and A. S. Kechris in [4], came from a paper of E. Michael [8]. In this paper he introduced the notion of a J-space, i.e. a topological space with the property that whenever {A,B} is a closed cover of X with A ∩ B compact, then A or B is compact. In terms of compactifications locally compact non-compact J-spaces are characterized by the property that their end-point compactification coincides with their one-point compactification (see [8, Proposition 6.2], [9, Theorem 6]). Recall that the Freudenthal or end-point compactification of a locally compact non-compact space X is the maximal zero-dimensional compactification εX of X. By zero-dimensional compactification of X we here mean a compactification Y of X such that Y \ X has a base of closed-open sets (see [7], [9]). The points of εX \ X are the ends of X. From the topological point of view locally compact spaces 2010 Mathematics Subject Classification. Primary 37B05; Secondary 54H20.