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 [Gro93]. 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 $d>1\slash 3$, 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 \.{Z}uk and Kotowski--Kotowski, who consider only groups in the $\Gamma (n,3k,d)$ model.