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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 Gcurvearrowright 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 SLm(ℤ)curvearrowright 𝕋m has geometric property (T) for m≥ 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.
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BibTeX
@misc{toyota2026warped,
title = {{Warped cones associated to isometric free actions do not have geometric property (T)}},
author = {Ryo Toyota},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24652},
eprint = {2608.24652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24652},
}
RIS
TY - UNPB
TI - Warped cones associated to isometric free actions do not have geometric property (T)
AU - Ryo Toyota
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.24652
UR - https://arxiv.org/abs/2608.24652
AB - Suppose that G is a finitely generated group, M is a compact Riemannian manifold and Gcurvearrowright 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 SL_m(ℤ)curvearrowright 𝕋^m has geometric property (T) for m≥ 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.
ER -