## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1227-1242.

Dates
Revised: 20 September 2004
Accepted: 21 September 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883466

Digital Object Identifier
doi:10.2140/gt.2004.8.1227

Mathematical Reviews number (MathSciNet)
MR2087082

Zentralblatt MATH identifier
1091.58021

#### Citation

Doyle, Peter G; Rossetti, Juan Pablo. Tetra and Didi, the cosmic spectral twins. Geom. Topol. 8 (2004), no. 3, 1227--1242. doi:10.2140/gt.2004.8.1227. https://projecteuclid.org/euclid.gt/1513883466

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