Morse subgroups and boundaries of random right-angled Coxeter groups

Abstract

We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erdős--Rényi model. We show that at densities below $\left(\sqrt{\frac{1}{2}}-ε\right)\sqrt{\frac{\log{n}}{n}}$ 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 $\left(\sqrt{\frac{1}{2}}+ε\right)\sqrt{\frac{\log{n}}{n}}$ 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 $(1-ε)\sqrt{\frac{\log{n}}{n}}$, $\square(Γ)$ 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.