Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups

TL;DR

This paper investigates the geometric properties of random right-angled Coxeter groups with Erdős–Rényi graph models, establishing new thresholds for connectivity and demonstrating typical cubical rigidity in these groups.

Contribution

It determines the connectivity threshold for square percolation and shows that a wide range of these groups have a unique cubical coarse median structure, revealing new rigidity phenomena.

Findings

Resolved a conjecture on connectivity threshold for square percolation.

Identified parameter ranges where random RACGs have cubical coarse median structures.

Showed cubical rigidity is common in non-hyperbolic random RACGs.

Abstract

We consider random right-angled Coxeter groups, $W_{\Gamma}$, whose presentation graph $\Gamma$ is taken to be an Erd\H{o}s--R\'enyi random graph, i.e., $\Gamma\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups. We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $\Gamma \sim \mathcal{G}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_{\Gamma}$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1/2$.