Tetra and Didi, the cosmic spectral twins

Abstract

We introduce a pair of isospectral but non-isometric compact flat 3–manifolds called $Tetra$ (a tetracosm) and $Didi$ (a didicosm). The closed geodesics of $Tetra$ and $Didi$ are very different. Where $Tetra$ has two quarter-twisting geodesics of the shortest length, $Didi$ has four half-twisting geodesics. Nevertheless, these spaces are isospectral. This isospectrality can be proven directly by matching eigenfunctions having the same eigenvalue. However, the real interest of this pair – and what led us to discover it – is the way isospectrality emerges from the Selberg trace formula, as the result of a delicate interplay between the lengths and twists of closed geodesics.