Homogeneous Actions on the Random Graph

TL;DR

This paper demonstrates that various complex group constructions, including free products and HNN extensions, can act faithfully and homogeneously on the Random Graph, revealing rich symmetry properties of these groups.

Contribution

It establishes new conditions under which complex groups admit faithful, homogeneous actions on the Random Graph, extending previous understanding of group actions on this structure.

Findings

Free products with an infinite factor admit faithful, homogeneous actions.

Certain HNN extensions and amalgamated free products also admit such actions.

Existence of dense free subgroups with all orbits infinite in the automorphism group of the Random Graph.

Abstract

We show that any free product of two countable groups, one of them being infinite, admits a faithful and homogeneous action on the Random Graph. We also show that a large class of HNN extensions or free products, amalgamated over a finite group, admit such an action and we extend our results to groups acting on trees. Finally, we show the ubiquity of finitely generated free dense subgroups of the automorphism group of the Random Graph whose action on it have all orbits infinite.