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Completely metrisable groups acting on trees. (English) Zbl 1268.20028

Summary: We consider actions of completely metrisable groups on simplicial trees in the context of the Bass-Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable group \(G\), namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation, \(G=A*_CB\), the amalgamated groups \(A\), \(B\) and \(C\) are open in \(G\).

MSC:

20E08 Groups acting on trees
03E15 Descriptive set theory
22D05 General properties and structure of locally compact groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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