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In this article we study the spectrum of totally geodesic surfaces of afinite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic3-manifolds that contain a totally geodesic surface, this spectrum determinesthe commensurability class. In addition, we show that any finite volumehyperbolic 3-manifold has many pairs of non-isometric finite covers withidentical spectra. Forgetting multiplicities, we can also construct pairs wherethe volume ratio is unbounded.

The genus spectrum of a hyperbolic 3-manifold