Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds

TL;DR

This paper constructs infinitely many pairs of 1-cusped hyperbolic 3-manifolds that are isospectral but not isometric, sharing spectral properties and using Sunada's method, with a uniqueness result for the figure-eight knot complement.

Contribution

It introduces a novel application of Sunada's method to construct infinitely many isospectral non-isometric 1-cusped hyperbolic 3-manifolds and proves a spectral uniqueness result for the figure-eight knot complement.

Findings

Existence of infinitely many isospectral non-isometric 1-cusped hyperbolic 3-manifolds.

These manifolds share the same spectrum, Eisenstein series, and complex length-spectra.

The figure-eight knot complement is uniquely determined by its spectrum among finite volume hyperbolic 3-manifolds.

Abstract

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length-spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.