Property (T) in random quotients of hyperbolic groups at densities above 1/3

Abstract

We study random quotients of a fixed non-elementary hyperbolic group in the Gromov density model. Let $G=\langle S\;\vert\; T\rangle $ be a finite presentation of a non-elementary hyperbolic group, and let $Ann_{l,\omega }(G)$ be the set of elements of norm between $l-\omega(l)$ and $l$ in $G$. A random quotient at density $d$ and length $\omega$-near $l$ is defined by killing a uniformly randomly chosen set of $\vert S_{l}(G)\vert ^{d}$ words in $Ann_{l,\omega (l)}(G)$, where $\omega (l) =o_{l}(l)$. We prove that for any d>1/3, such a quotient has Property (T) with probability tending to $1$ as $l$ tends to infinity. This result answers a question of Gromov--Ollivier and strengthens a theorem of \.{Z}uk (c.f Kotowski--Kotowski).