Property FA for random $\mathrm{\ell }$-gonal groups

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