Fundamental groups of clique complexes of random graphs

Abstract

We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs Γ . We establish thresholds for a number of properties of fundamental groups of these complexes XΓ . In particular, if p=nα , then we show that gdim(π1(XΓ))=cd(π1(XΓ))=1ifα<−12,gdim(π1(XΓ))=cd(π1(XΓ))=2if−12−13 . We prove that for −1130