#### Volume 8, issue 3 (2004)

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Tetra and Didi, the cosmic spectral twins

### Peter G Doyle and Juan Pablo Rossetti

Geometry & Topology 8 (2004) 1227–1242
 arXiv: math.DG/0407422
##### 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.

##### Keywords
flat structure, 3–manifold, platycosm, Laplace spectrum, isospectral, Selberg trace formula, closed geodesic
##### Mathematical Subject Classification 2000
Primary: 57M50, 58J53
Secondary: 11F72, 53C22