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Groups, Geometry, and Dynamics

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Volume 15, Issue 1, 2021, pp. 1–34
DOI: 10.4171/GGD/589

Published online: 2020-12-16

Homogeneous actions on the random graph

Pierre Fima[1], Soyoung Moon[2] and Yves Stalder[3]

(1) Université de Paris, Sorbonne Université, France
(2) Université de Bourgogne, Dijon, France
(3) Université Clermont Auvergne, Clermont-Ferrand, and Université Blaise Pascal, Aubière, France

We show that any free product of two (non-trivial) 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.

Keywords: Homogeneous actions, random graph, free groups, groups acting on trees, Baire category theorem

Fima Pierre, Moon Soyoung, Stalder Yves: Homogeneous actions on the random graph. Groups Geom. Dyn. 15 (2021), 1-34. doi: 10.4171/GGD/589