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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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