Non-geometric property (T) of warped cones

Abstract

In this paper, we study the geometric property (T) for discretized warped cones of actions on a compact Lie group $M$ by its finitely generated subgroups. We show that if a subgroup $G$ is dense in $M$, then the associated discretized warped cone ${⨆}_{n}M\times \{t(n)\}$ does not have geometric property (T) for any sequence of positive numbers $\{t(n){\}}_{n\in \mathbb{N}}$ converging to $\infty$. This result applies to certain ergodic actions of groups with property (T), for example, the action of $\mathrm{SO}(d,\mathbb{Z}[\frac{1}{5}\left]\right)$ on $\mathrm{SO}(d)$ with $d\ge 5$. As an application, we obtain new examples of expanders without geometric property (T), including certain superexpanders.