Sarnak’s conjecture on the closure of bass notes for finite-volume hyperbolic 3-orbifolds

Determine whether the closure of the set of first Laplace eigenvalues of finite-volume hyperbolic 3-orbifolds equals [0,1] ∪ E, where E is an upper-bounded, discrete, and infinite subset of (1,∞).

Background

The authors define Bass(Hyp^3_{f.v.}) as the set of first Laplace eigenvalues over all finite-volume hyperbolic 3-orbifolds and Bass(Hyp^3_{a.}) for arithmetic ones. By Mostow–Prasad rigidity, these sets are countable.

Sarnak’s conjecture predicts that the closure of Bass(Hyp^3_{f.v.}) fills the entire interval [0,1] and consists of a discrete, upper-bounded infinite set above 1. The authors prove a first step: [0, λ0(Γ∞\H^3)] is contained in the closure of Bass(Hyp^3_{a.}), hence also in the closure of Bass(Hyp^3_{f.v.}).