Morse subgroups and boundaries of random right-angled Coxeter groups

We study Morse subgroups and boundaries of random right-angled Coxeter groups in the Erdős–Rényi model. show that at densities below $$\left( \sqrt{\frac{1}{2}}-\epsilon \right) \sqrt{\frac{\log {n}}{n}}$$ almost surely have hyperbolic surface subgroups. This implies their contain embedded circles. Further, above \sqrt{\frac{1}{2}}+\epsilon we that, surely, special a group are virtually free. also apply these methods to for graph $$\Gamma $$ $$(1-\epsilon )\sqrt{\frac{\log , $$\square (\Gamma )$$ contains an isolated vertex. shows, particular, is not quasi-isometric Artin