Virtually fibering random right-angled Coxeter groups

Abstract

We show that the Right-Angled Coxeter group $C=C(G)$ associated to a random graph $G\sim \mathcal{G}(n,p)$ with $\frac{\log n+\log \log n+\ddot{I}‰(1)}{n}\le p<1-\ddot{I}‰(n^{-2})$ virtually algebraically fibers. This means that $C$ has a finite index subgroup $C^{\prime }$ and a finitely generated normal subgroup $N\subset C^{\prime }$ such that $C^{\prime }/N\cong \mathbb{Z}$. 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.