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

TL;DR

This paper investigates Property (T) in various models of random groups, establishing bounds and probabilistic thresholds, notably showing that for densities above 1/3, random groups almost surely have Property (T) as parameters grow.

Contribution

It provides new bounds for Property (T) in the fixed-parameter model and extends known results to higher densities and larger parameters, strengthening previous theorems.

Findings

For $k$ tending to infinity, bounds for Property (T) are established.

Random groups with $d > 1/3$ have Property (T) with high probability as $k$ increases.

Results extend previous work to broader models and higher densities.

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.