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Spherical density for real semisimple groups
Abstract
Let G be a connected semisimple real Lie group with finite center and let (ΓN) be a tower of cocompact lattices. We prove that the weak injective radius property implies the spherical density hypothesis with the same parameter. Together with the converse theorem of Golubev and Kamber, this gives an equivalence in the cocompact Archimedean setting. The proof uses a uniform spherical-mass estimate near spectral walls and a positive pre-trace kernel, but no tempered-spectrum estimate or a priori spectral gap.
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BibTeX
@misc{lutsko2026spherical,
title = {{Spherical density for real semisimple groups}},
author = {Christopher Lutsko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00298},
eprint = {2610.00298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00298},
}
RIS
TY - UNPB
TI - Spherical density for real semisimple groups
AU - Christopher Lutsko
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00298
UR - https://arxiv.org/abs/2610.00298
AB - Let G be a connected semisimple real Lie group with finite center and let (Γ_N) be a tower of cocompact lattices. We prove that the weak injective radius property implies the spherical density hypothesis with the same parameter. Together with the converse theorem of Golubev and Kamber, this gives an equivalence in the cocompact Archimedean setting. The proof uses a uniform spherical-mass estimate near spectral walls and a positive pre-trace kernel, but no tempered-spectrum estimate or a priori spectral gap.
ER -