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=\operatorname{Iso}(X,d)) on a locally compact metric space ((X,d)) that possesses exactly one end. The notion of pseudo‑components, introduced by Gao and Kechris, plays a central role. A pseudo‑component is a maximal subset that is internally distance‑connected and whose points stay a positive distance apart from points in other components. Using this decomposition, the author proves three main facts. First, (X) can be partitioned into only finitely many pseudo‑components. Second, among these components precisely one is non‑compact; all the others are compact. Third, the action of (G) is proper on the unique non‑compact pseudo‑component, meaning that for any compact subset (C) of this component the set ({g\in G\mid gC\cap C\neq\varnothing}) is compact in (G). The complement of the non‑compact component is a compact subset of (X), but the group need not act properly on it; explicit examples are constructed where the action fails to be proper on the compact part.

The proof proceeds by exploiting the “one‑end’’ hypothesis: for any compact (K\subset X) the complement (X\setminus K) is connected, which forces all points that escape every compact set to lie in a single distance‑connected cluster. This yields the uniqueness of the non‑compact pseudo‑component. Properness on this component follows from standard arguments about displacement functions and the closedness of stabilisers in a locally compact group of isometries. The failure of properness on the compact remainder is demonstrated by a space consisting of a non‑compact “cylindrical’’ part attached to several compact pieces; an isometry that translates along the cylinder while permuting the compact pieces produces non‑compact stabilisers for compact subsets of the remainder.

The results extend classical theorems stating that isometry groups act properly on complete, locally compact, and globally compact spaces. Here the authors show that when the space has a single end, properness is retained only on the unique non‑compact pseudo‑component. The paper concludes with a discussion of possible generalisations: spaces with multiple ends, non‑locally compact metric spaces, and the role of pseudo‑components in broader dynamical studies of transformation groups. Overall, the work provides a clear structural picture of how the topology of a space with one end controls the dynamical behaviour of its isometry group.