Sarnak's spectral gap question
Spectral Theory
2022-10-26 v1 Geometric Topology
Abstract
We answer in the affirmative a question of Sarnak's from 2007, confirming that the Patterson-Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian packing. Thus the latter has a maximal spectral gap. We prove further restrictions on the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere packings.
Cite
@article{arxiv.2210.13969,
title = {Sarnak's spectral gap question},
author = {Dubi Kelmer and Alex Kontorovich and Christopher Lutsko},
journal= {arXiv preprint arXiv:2210.13969},
year = {2022}
}
Comments
8 pages, 4 figures