Property (T) in density-type models of random groups

Abstract

We study property (T) in the $\Gamma (n,k,d)$ model of random groups: as $k$ tends to infinity this gives the Gromov density model, introduced in 1993. We provide bounds for property (T) in the $k$-angular model of random groups, i.e., the $\Gamma (n,k,d)$ model, where $k$ is fixed and $n$ tends to infinity. We also prove that for $d>1/3$, a random group in the $\Gamma (n,k,d)$ model has property (T) with probability tending to $1$ as $k$ tends to infinity, strengthening the results of Żuk and Kotowski–Kotowski, who consider only groups in the $\Gamma (n,3k,d)$ model.