Paper published by Geometry and Topology

From: Geometry and Topology Journal (gt_at_maths.warwick.ac.uk)
Date: 09/22/04 

Date: 22 Sep 2004 09:54:45 -0400

The following paper has been published:

Geometry and Topology, Volume 8 (2004) Paper no. 33, pages 1227--1242

URL:
http://www.maths.warwick.ac.uk/gt/GTVol8/paper33.abs.html

Title:
Tetra and Didi, the cosmic spectral twins

Author(s):
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.

AMS Classification Numbers. Primary: 57M50, 11F72
Secondary: 11M38, 57N15

Keywords:
Flat structure, 3-manifold, platycosm, Laplace spectrum, isospectral,
Selberg trace formula

Received: 17 July 2003
Revised: 20 September 2004
Accepted: 21 September 2004
Published: 22 September 2004

Proposed: Walter Neumann
Seconded: Benson Farb, Steven Ferry

Author(s) address(es):
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