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We show that any free product of two countable groups, one of them beinginfinite, admits a faithful and homogeneous action on the Random Graph. We alsoshow that a large class of HNN extensions or free products, amalgamated over afinite group, admit such an action and we extend our results to groups actingon trees. Finally, we show the ubiquity of finitely generated free densesubgroups of the automorphism group of the Random Graph whose action on it haveall orbits infinite.

Homogeneous Actions on the Random Graph