Property FA for random $$\ell $$-gonal groups

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