Virtually fibering random right-angled Coxeter groups

TL;DR

This paper proves that random right-angled Coxeter groups associated with certain Erdős–Rényi graphs almost surely have a finite index subgroup with a normal subgroup such that the quotient is infinite cyclic, extending previous work.

Contribution

It establishes that under specific probabilistic conditions, these groups virtually algebraically fiber, improving understanding of their algebraic structure in random graph models.

Findings

Random right-angled Coxeter groups virtually algebraically fiber under specified conditions.

The result is essentially optimal given the probabilistic thresholds.

The work extends and builds upon previous research by Jankiewicz, Norin, and Wise.

Abstract

We show that the Right-Angled Coxeter group $C=C(G)$ associated to a random graph $G\sim \mathcal{G}(n,p)$ with $\frac{\log n + \log\log n + \omega(1)}{n} \leq p < 1- \omega(n^{-2})$ virtually algebraically fibers. This means that $C$ has a finite index subgroup $C'$ and a finitely generated normal subgroup $N\subset C'$ such that $C'/N \cong \mathbb{Z}$. We also obtain the corresponding hitting time statements, more precisely, we show that as soon as $G$ has minimum degree at least 2 and as long as it is not the complete graph, then $C(G)$ virtually algebraically fibers. The result builds upon the work of Jankiewicz, Norin, and Wise and it is essentially best possible.