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 $$\left( \sqrt{\frac{1}{2}}-\epsilon \right) \sqrt{\frac{\log {n}}{n}}$$ random right-angled Coxeter groups almost surely have Morse hyperbolic surface subgroups. This implies their Morse boundaries contain embedded circles. 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. This shows, in particular, that at densities below $$(1-\epsilon )\sqrt{\frac{\log {n}}{n}}$$ a random right-angled Coxeter group is almost surely not quasi-isometric to a right-angled Artin group