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< 12 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-\'Swiatkowski and Dahmani-Guirardel-Przytycki.
Clément et al. (2025) studied this question.