Constructing multi-cusped hyperbolic manifolds that are isospectral and not isometric

TL;DR

This paper extends the construction of hyperbolic 3-manifolds that are isospectral but not isometric from one cusp to multiple cusps, using advanced mathematical techniques to produce arbitrarily many cusped examples.

Contribution

It introduces a method to construct multi-cusped hyperbolic 3-manifolds that are isospectral but not isometric, generalizing previous single-cusp results.

Findings

Constructed hyperbolic 3-manifolds with multiple cusps that are isospectral and non-isometric.

Manifolds share the same Eisenstein series and spectrum.

The construction employs Sunada's method and the Strong Approximation Theorem.

Abstract

In a recent paper Garoufalidis and Reid constructed pairs of 1-cusped hyperbolic 3-manifolds which are isospectral but not isometric. In this paper we extend this work to the multi-cusped setting by constructing isospectral but not isometric hyperbolic 3-manifolds with arbitrarily many cusps. The manifolds we construct have the same Eisenstein series, the same infinite discrete spectrum and the same complex length spectrum. Our construction makes crucial use of Sunada's method and the Strong Approximation Theorem of Nori and Weisfeiler.