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On the group of isometries on a locally compact metric space. (English) Zbl 1012.54046
Let \((X,d)\) be a locally compact metric space and let \(I(X,d)\) be its group of isometries endowed with the topology of pointwise convergence. Let \(\Sigma(X)\) denote the space of the connected components endowed with the quotient topology. The authors prove the following statements concerning the group \(I(X,d)\) and its action on the space \(X\): (1) If \(\Sigma(X)\) is not quasicompact, then \(I(X,d)\) need not be locally compact, nor act properly on \(X\). (2) If \(\Sigma(X)\) is quasicompact then (a) \(I(X,d)\) is locally compact, (b) the action of it on \(X\) is not always proper, and (c) the action of it is proper if \(X\) is connected.

54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics
54E15 Uniform structures and generalizations
54H15 Transformation groups and semigroups (topological aspects)
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