Mathematical Research Letters

Volume 21 (2014)

Number 1

The genus spectrum of a hyperbolic 3-manifold

Pages: 169 – 185



D. B. McReynolds (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

A. W. Reid (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)


In this paper, we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold has many pairs of non-isometric finite covers with identical spectra. Forgetting multiplicities, we can also construct pairs where the volume ratio is unbounded.

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