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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 \(P\).
54E45 Compact (locally compact) metric spaces
54H11 Topological groups (topological aspects)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Full Text: DOI arXiv
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