Sarnak's spectral gap question

TL;DR

This paper confirms Sarnak's 2007 question by proving the uniqueness of the Patterson-Sullivan eigenfunction for Apollonian packings, establishing a maximal spectral gap and exploring spectral restrictions on related manifolds.

Contribution

It provides a definitive answer to Sarnak's spectral gap question and extends spectral analysis to a broad class of Kleinian sphere packing manifolds.

Findings

The Patterson-Sullivan eigenfunction is unique and square-integrable.

The Apollonian packing has a maximal spectral gap.

Spectral restrictions apply to manifolds from Kleinian sphere packings.

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.