Apollonian random manifolds and their bass notes

TL;DR

This paper investigates the spectral properties of random hyperbolic 3-orbifolds related to Apollonian groups, providing explicit spectral gaps and exploring the bass note spectrum of these geometric structures.

Contribution

It introduces models of random hyperbolic 3-orbifolds linked to Apollonian groups and determines their spectral gaps, advancing understanding of their geometric spectra.

Findings

Explicit spectral gaps for the studied orbifolds.

Identification of the bass note spectrum in hyperbolic 3-orbifolds.

Insights into the spectral distribution related to Apollonian structures.

Abstract

We study the spectrum of the Laplacian on two models of random hyperbolic 3-orbifolds, related to the Apollonian group and the super Apollonian group. We determine explicit spectral gaps for these random orbifolds. Moreover, we use our model to investigate the bass note spectrum of the set of hyperbolic 3-orbifolds.