Property FA for random $\ell$-gonal groups

TL;DR

This paper investigates the phase transition in random $ ext{ell}$-gonal groups, identifying a critical density at which the groups acquire property FA and analyzing the topological and algebraic consequences of this threshold.

Contribution

It establishes a double threshold near density $d=1/\ell$ for property FA in random $ ext{ell}$-gonal groups and explores related topological and algebraic properties.

Findings

Threshold at $d=1/\ell$ for property FA

Boundary homeomorphic to Menger sponge for $d>1/\ell$

Threshold for finiteness of $\mathrm{Out}(G)$ at $d=1/\ell$

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-\'Swi\k{a}tkowski and Dahmani-Guirardel-Przytycki.