2021/08/22 by Tim Susse, Susse, Tim
Computer Science · Mathematics · #05C80 #20F36 #20F55 #20F65 #20F69 #57M07 #57M15 #60B99 #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2108.09824
openalex publication_date 2021/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erdős--Rényi model. We show that at densities below (√((1)/(2))-ε)√\fraclognn random right-angled Coxeter groups almost surely have Morse hyperbolic surface subgroups. This implies their Morse boundaries contain embedded circles and they cannot be quasi-isometric to a right-angled Artin group. Further, at densities above (√((1)/(2))+ε)√\fraclognn we show that, almost surely, the hyperbolic Morse special subgroups of a random right-angled Coxeter group are virtually free. We also apply these methods to show that for a random graph Γ at densities below (1-ε)√\fraclognn, \square(Γ) almost surely contains an isolated vertex. As a consequence, this provides infinitely many examples of right-angled Coxeter groups with no one-ended hyperbolic Morse special subgroups that are not quasi-isometric to a right-angled Artin group.