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The group of isometries of a locally compact metric space with one end. (English) Zbl 1207.54038
Recall that the Freudenthal or end-point compactification of a locally compact space $$X$$ is the maximal compactification $$\varepsilon X$$ with the property that the set $$\varepsilon X\setminus X$$ of end-points is zero-dimensional. Let $$(X,d)$$ be a locally compact metric space and let $$G$$ be its group of isometries with the topology of pointwise convergence. For $$x\in X$$ let $$L(x)=\{\lim g_i(x):\{g_i\}$$ is a net in $$G$$ with no cluster point$$\}$$. An action $$(g,x)\mapsto g(x)$$ given by an isometry $$g$$ is proper if $$L(x)=\emptyset$$ for all $$x\in X$$. The value $$\rho(x)=\sup\{r>0:B(x,r)$$ has compact closure$$\}$$ is called a radius of compactness of $$x\in X$$. Let $$\mathcal R^*$$ be the transitive closure of the relation $$\mathcal R=\{(x,y)\in X\times X:d(x,y)<\rho(x)\}$$ and let $$\mathcal E$$ be the equivalence relation defined by $$x\mathcal Ey$$ if and only if $$x=y$$ or $$(x\mathcal R^*y$$ or $$y\mathcal R^*x)$$. The $$\mathcal E$$-equivalence class of $$x$$ is called a pseudo-component of $$x$$. The author proves that if $$(X,d)$$ is a locally compact metric space with one end, then $$X$$ has finitely many pseudo-components, exactly one pseudo-component $$P$$ is not compact, $$X\setminus P$$ is a compact subset of $$X$$, and $$G$$ acts properly on $$P$$.
##### MSC:
 54E45 Compact (locally compact) metric spaces 54H11 Topological groups (topological aspects) 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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