Amenable, transitive and faithful actions of groups acting on trees
Pierre Fima (IMJ)

TL;DR
This paper investigates conditions under which groups acting on trees, such as amalgamated free products and HNN-extensions, admit amenable, transitive, and faithful actions on countable sets, extending known results using Baire category arguments.
Contribution
It establishes new criteria for groups acting on trees to have amenable, transitive, and faithful actions, broadening understanding of group actions in this context.
Findings
Existence of such actions if initial groups have amenable, almost free actions with infinite orbits
Extension of results to groups acting on trees
Use of Baire category Theorem in the analysis
Abstract
We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topology and Set Theory · Geometric and Algebraic Topology
