New York Journal of Mathematics
Volume 14 (2008) 193-204


Peter G. Doyle and Juan Pablo Rossetti

Isospectral hyperbolic surfaces have matching geodesics

Published: June 5, 2008
Keywords: Isospectrality, closed geodesic, holonomy, almost conjugate, hyperbolic surface, Selberg trace formula, flat orbifold, prime geodesic theorem
Subject: Primary: 58J53; Secondary: 11F72, 20F67, 53C22

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.


Partially supported by DFG Sonderforschungsbereich 647, Humboldt University, Berlin.

Author information

Peter G. Doyle:
Dartmouth College

Juan Pablo Rossetti:
FaMAF-CIEM, Univ. Nac. Córdoba