Resumen de Sharp vanishing thresholds for cohomology of random flag complexes

Matthew Kahle

• For every k=1 , the k -th cohomology group H k (X,Q) of the random flag complex X~X(n,p) passes through two phase transitions: one where it appears and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group p 1 (X) to have Kazhdan�s property~(T). Combining with earlier results, we obtain as a corollary that for every k=3 , there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of k -dimensional spheres.