Completely Metrisable Groups Acting on Trees

TL;DR

This paper studies actions of completely metrisable groups on trees, characterizing when the amplitude function is continuous and showing that certain subgroups in free product decompositions are open under mild conditions.

Contribution

It provides a new characterization of amplitude function continuity and proves openness of subgroups in free product decompositions for a broad class of groups.

Findings

Amplitude function continuity characterized for group actions on trees

Subgroups in free product decompositions are open under mild conditions

Conditions on generating elements relate to subgroup discreteness and density

Abstract

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, $G=A\ast_CB$, the amalgamated groups $A$, $B$ and $C$ are open in $G$.