Spectral gaps of closed hyperbolic 3-manifolds approaching 1

Establish the existence of sequences of closed hyperbolic 3-manifolds whose volumes tend to infinity and whose first nonzero Laplace eigenvalues (spectral gaps) converge to 1, the bottom of the spectrum of hyperbolic 3-space H^3.

Background

The paper studies spectral gaps for random hyperbolic 3-orbifolds and related constructions, providing explicit gaps and connections to the bass note spectrum. In dimension two, strong results exist on constructing large-area surfaces with near-optimal spectral gaps, motivating analogous questions in dimension three.

The cited conjecture asks for closed hyperbolic 3-manifolds with spectral gaps tending to that of H^3 (which equals 1), as the volume grows. The authors’ results give sequences whose first eigenvalue approaches λ0(Γ∞\H^3), a value strictly below 1, and therefore represent progress but do not resolve this conjecture.