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Tetra and Didi, the cosmic spectral twins

Peter G Doyle·Juan Pablo Rossetti
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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.

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Peter G Doyle, Juan Pablo Rossetti. 2006-05-31. Tetra and Didi, the cosmic spectral twins. https://doi.org/10.2140/gt.2004.8.1227

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