A threshold for relative hyperbolicity in random right-angled Coxeter groups

2024/07/17 by Behrstock, Jason, Ciceksiz, Recep Altar, Falgas-Ravry, Victor
#05C80 20F55 20F67 20F65 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2407.12959

Abstract

We consider the random right-angled Coxeter group W_Γ whose presentation graph Γ∼ G_n,p is an Erd\H os--Rényi random graph on n vertices with edge probability p=p(n). We establish that p=1/√(n) is a threshold for relative hyperbolicity of the random group W_Γ. As a key step in the proof, we determine the minimal number of pairs of generators that must commute in a right-angled Coxeter group which is not relatively hyperbolic, a result which is of independent interest. We also show that there is an interval of edge probabilities of width Ω(1/√(n)) in which the random right-angled Coxeter group has precisely cubic divergence. This interval is between the thresholds for relative hyperbolicity (whence exponential divergence) and quadratic divergence. Moreover, a simple random walk on any Cayley graph of the random right-angled Coxeter group for p in this interval satisfies a central limit theorem.

Related

• Random Coxeter Groups
• Square percolation and the threshold for quadratic divergence in random right-angled Coxeter groups
• Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups
• Virtually fibering random right-angled Coxeter groups
• Morse subgroups and boundaries of random right-angled Coxeter groups