Tetra and Didi, the cosmic spectral twins

Doyle, Peter G; Rossetti, Juan Pablo

doi:10.2140/gt.2004.8.1227

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Tetra and Didi, the cosmic spectral twins

Peter G Doyle and Juan Pablo Rossetti

Geometry & Topology 8 (2004) 1227–1242

DOI: 10.2140/gt.2004.8.1227

arXiv: math.DG/0407422

Abstract

We introduce a pair of isospectral but non-isometric compact flat 3–manifolds called $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Tetra</mo></math>$ (a tetracosm) and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Didi</mo></math>$ (a didicosm). The closed geodesics of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Tetra</mo></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Didi</mo></math>$ are very different. Where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Tetra</mo></math>$ has two quarter-twisting geodesics of the shortest length, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Didi</mo></math>$ 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

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Publication

Received: 17 July 2003

Revised: 20 September 2004

Accepted: 21 September 2004

Published: 22 September 2004

Proposed: Walter Neumann

Seconded: Benson Farb, Steven Ferry

Authors

Peter G Doyle
Department of Mathematics Dartmouth College Hanover New Hampshire 03755-3551 USA

Juan Pablo Rossetti
FaMAF – Ciem Universidad Nacional de Córdoba Argentina

Volume 8, issue 3 (2004)