Amenable actions of amalgamated free products

TL;DR

This paper demonstrates that certain amalgamated free products, including some surface groups, can act amenably, faithfully, and transitively on countable sets, extending to their finite index subgroups.

Contribution

It establishes the existence of amenable, faithful, transitive actions for amalgamated free products of free groups over cyclic subgroups, including surface groups.

Findings

Amalgamated free products of two free groups of rank two over a cyclic subgroup admit such actions.

Finite index subgroups of these amalgamated products also admit such actions.

Surface groups and fundamental groups of surface bundles over S^1 are included in these results.

Abstract

We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over $\mathbb{S}^1$.