Property FA for random $\ell$-gonal groups
Emily Clement, John M. Mackay
TL;DR
This work analyzes random binomial $\ell$-gonal groups in the sense of Gromov and related models, establishing a sharp double threshold for Serre's property FA near density $d=\frac{1}{\ell}$ and delineating three phases: a free phase at low density, a non-free non-FA phase at intermediate density (where $G$ splits as $H * \mathbb{Z} * \mathbb{Z}$), and an FA phase at high density. It shows that for densities $\frac{1}{\ell} < d < \frac{1}{2}$ the boundary $\partial_\infty G$ is a Menger sponge, and identifies a density range where $\mathrm{Out}(G)$ transitions from infinite to finite, with corresponding finiteness for Hom$(G,\Gamma)$ in torsion-free hyperbolic $\Gamma$. The paper also discusses the relationship between FA and Kazhdan's property (T), providing ranges where FA holds but (T) may fail, and develops a positive $\ell$-gonal model to prove FA via relator-positivity arguments, connecting to Gromov-density methods. Methodologically, the authors combine triangular-model techniques (deletion arguments, hypergraph analyses) with quotient-transfer principles to the Gromov density framework, yielding a detailed map of how density and relator length control group splittings, boundary topology, and outer automorphism groups.
Abstract
In the binomial $\ell$-gonal model for random groups, where the random relations all have fixed length $\ell\geq 3$ 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 $\ell \geq 4$. Our methods are inspired by work of Antoniuk-Luczak-Świątkowski and Dahmani-Guirardel-Przytycki.