CONSTRUCTING MULTICUSPED HYPERBOLIC MANIFOLDS THAT ARE ISOSPECTRAL AND NOT ISOMETRIC

Abstract

In a recent paper Garoufalidis and Reid constructed pairs of $1$-cusped hyperbolic $3$-manifolds which are isospectral but not isometric. We extend this work to the multicusped 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.