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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References:

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