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 pseudo-components introduced by S. Gao and A. S. Kechris we show that $X$ has only finitely many pseudo-components of which exactly one is not compact and $G$ acts properly on. The complement of the non-compact component is a compact subset of $X$ and $G$ may fail to act properly on it.

💡 Research Summary

The paper investigates the natural evaluation action of the isometry group (G) on a locally compact metric space (X) that possesses exactly one end. The author builds on the notion of pseudo‑components introduced by S. Gao and A. S. Kechris, and on the classical concept of a J‑space (a topological space in which any closed cover ({A,B}) with compact intersection forces at least one of the sets to be compact). For locally compact non‑compact J‑spaces, the Freudenthal end compactification coincides with the one‑point compactification, a fact that underlies much of the analysis.

A pseudo‑component is defined via the “radius of compactness’’ (\rho(x)=\sup{r>0\mid \overline{B(x,r)}\text{ is compact}}). Two points are related by a directed graph (R) when their distance is smaller than (\rho) of the first point; the transitive closure (R^{*}) together with its symmetric counterpart yields an equivalence relation (E). The equivalence class (C_{x}) is the pseudo‑component of (x). Important properties are: each pseudo‑component is a closed‑open subset of (X); (\rho) is invariant under isometries, so (gC_{x}=C_{gx}); and the family of pseudo‑components forms a closed‑open partition of (X).

The main result (Theorem 1.1) consists of two parts. (i) (X) has only finitely many pseudo‑components, exactly one of which is non‑compact; denote this component by (P). Moreover, (P) is invariant under the whole group (G) and (G) is locally compact. (ii) (G) acts properly on the non‑compact component (P). The complement (X\setminus P) is compact, but the action of (G) on this compact part need not be proper; indeed, examples are given where isotropy subgroups are non‑compact.

The proof proceeds as follows. Lemma 1.2 shows that any partition of a non‑compact J‑space into closed‑open non‑empty sets must consist of finitely many pieces, exactly one of which is non‑compact. Applying this lemma to the family of pseudo‑components yields the finiteness statement. Proposition 1.3 establishes that a metrizable locally compact J‑space is second‑countable, which allows the use of countable bases throughout the argument.

A crucial technical tool is Proposition 1.4 (originally Theorem 1.3 of