Geometry & Topology, Vol. 8 (2004) Paper no. 33, pages 1227--1242.

Tetra and Didi, the cosmic spectral twins

Peter G Doyle, Juan Pablo Rossetti

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

AMS subject classification. Primary: 57M50, 11F72. Secondary: 11M38, 57N15.

DOI: 10.2140/gt.2004.8.1227

E-print: arXiv:math.DG/0407422

Submitted to GT on 17 July 2003. (Revised 20 September 2004.) Paper accepted 21 September 2004. Paper published 22 September 2004.

Notes on file formats

Peter G Doyle, Juan Pablo Rossetti

Department of Mathematics, Dartmouth College
Hanover, NH 03755-3551, USA
and
FaMAF -- Ciem, Universidad Nacional de Cordoba, Argentina

Email: doyle@math.dartmouth.edu, rossetti@mate.uncor.edu

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