2025/05/12 by Emily Clément, Clement, Emily, John M. Mackay +1
Mathematics · #20E08 #20F05 #20F65 #20F67 #20P05 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2505.07424
openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the binomial ℓ-gonal model for random groups, where the random relations all have fixed length ℓ≥ 3 and the number of generators goes to infinity, we establish a double threshold near density d=(1)/(ℓ) where the group goes from being free to having Serre's property FA. As a consequence, random ℓ-gonal groups at densities (1)/(ℓ) < d< (1)/(2) have boundaries homeomorphic to the Menger sponge, and (1)/(ℓ) is also the threshold for finiteness of Out(G). We also see that the thresholds for property FA and Kazhdan's property (T) differ when ℓ ≥ 4. Our methods are inspired by work of Antoniuk-Luczak-Świątkowski and Dahmani-Guirardel-Przytycki.