an:05835568
Zbl 1207.54038
Manoussos, Antonios
The group of isometries of a locally compact metric space with one end
EN
Topology Appl. 157, No. 18, 2876-2879 (2010).
00270015
2010
j
54E45 54H11 54D35
proper action; pseudo-component; Freudenthal compactification; end-point compactification; ends; \(J\)-space
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~\(P\).
Miroslav Repick?? (Ko??ice)