Non-geometric property (T) of warped cones

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 \bigsqcup_{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}\bigl(d,\mathbb{Z}\bigl[\frac{1}{5}\bigr]\bigr) on \mathrm{SO}(d) with d\geq 5 . As an application, we obtain new examples of expanders without geometric property (T), including certain superexpanders.