Property FA for random -gonal groups

In the binomial -gonal model for random groups, where the random relations all have fixed length ≥ 3 and the number of generators goes to infinity, we establish a double threshold near density d=1/ where the group goes from being free to having Serre's property FA. As a consequence, random -gonal groups at densities 1/ < d< 1/2 have boundaries homeomorphic to the Menger sponge, and 1/ is also the threshold for finiteness of Out(G). We also see that the thresholds for property FA and Kazhdan's property (T) differ when ≥ 4. Our methods are inspired by work of Antoniuk-Luczak-\'Swi{a}tkowski and Dahmani-Guirardel-Przytycki.