Warped cones associated to isometric free actions do not have geometric property (T)

Abstract

Suppose that $G$ is a finitely generated group, $M$ is a compact Riemannian manifold and $G\curvearrowright M$ is a free and isometric action. We prove that the associated discretized warped cone does not have geometric property (T). In contrast, we show that the discretized warped cone associated to the natural action $S{L}_m(\mathbb{Z})\curvearrowright {\mathbb{T}}^m$ has geometric property (T) for $m\ge 3$. We also prove that, for free probability-measure-preserving Lipschitz actions on compact Riemannian manifolds, geometric property (T) of the discretized warped cone implies Kazhdan's property (T) of the acting group.

An example of action whose warped cone has geometric property (T) was added