Morse subgroups and boundaries of random right-angled Coxeter groups

TL;DR

This paper investigates the properties of Morse subgroups and boundaries in random right-angled Coxeter groups, revealing phase transitions in subgroup structures at different graph densities.

Contribution

It establishes new density thresholds for the existence of Morse hyperbolic surface subgroups and virtually free Morse subgroups in random right-angled Coxeter groups.

Findings

Below a certain density, groups almost surely contain Morse hyperbolic surface subgroups.

Above a certain density, hyperbolic Morse special subgroups are virtually free.

At lower densities, random graphs almost surely have isolated vertices, affecting subgroup structures.

Abstract

We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erd\H{o}s--R\'enyi model. We show that at densities below $\left(\sqrt{\frac{1}{2}}-\epsilon\right)\sqrt{\frac{\log{n}}{n}}$ 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 $\left(\sqrt{\frac{1}{2}}+\epsilon\right)\sqrt{\frac{\log{n}}{n}}$ 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 $\Gamma$ at densities below $(1-\epsilon)\sqrt{\frac{\log{n}}{n}}$, $\square(\Gamma)$ 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.