Property FA for random $\ell$-gonal groups

Abstract

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

v1: 15 pages; v2: 16 pages, minor updates