Property (T) for random groups in the density model with $d>\frac{1}{4}$

Abstract

We prove that random groups in the Gromov density model (under a certain divisibility condition) have property (T) asymptotically almost surely for any density $d>\frac{1}{4}$. This established $\frac{1}{4}$ as the threshold for property (T), since it was shown by Ashcroft that they do not have property (T) for $d<\frac{1}{4}$.