Small volume isospectral, non-isometric, hyperbolic 2-orbifolds

Abstract

0. If S is a complete orientable hyperbolic 2-orbifold of finite volume then S = 1-12/I , where F is a Fuchsian group. The geodesic length spectrum of S is the sequence of translation lengths of conjugacy classes of primitive hyperbolic elements of 1,, counted with multiplicities. The spectrum of the Laplace-Beltrami operator determines the length spectrum together with the signature of the group F and vice versa (see e.g. [5]). Two groups whose spectra agree in either sense are called isospectral. Note that two groups whose spectra agree except for a finite number must actually be isospectral. In particular, the spectrum determines the signature of the group. It is a classical problem as to how far the spectrum determines the hyperbolic structure of S. "Generically" the spectrum does determine the hyperbolic structure up to isometry [12] but Vigneras [10] showed, using arithmetic methods, which are also applicable in higher dimensions, that isospectral non-isometric hyperbolic 2-manifolds exist. Subsequently, using a general technique of Sunada [9] which reduces the isospectrality conditions to certain properties of finite groups, it was shown [2], [1] that there exist pairs of compact qs0spectral n0n-isometric hyperbolic 2-manifolds of every genus g __> 4. At the other extreme, orbifolds corresponding to triangle groups are rigid, and so, once the signature is determined, are necessarily isometric. More importantly, Haas [4] has shown that hyperbolic isospectral once-punctured tori are necessarily isometric. Furthermore, Haas' argument can be adapted to give the same result for those orbifolds which are tori with one cone point. In this note, we return to Vigneras' method and apply it to certain examples which arose in the detailed analysis of all arithmetic Fuchsian groups of signature (0; 2, 2, 2, e; 0) where e is odd [6], to show