Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron |
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Authors: | Dubravko Ivan
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Institution: | University of Oklahoma, Mathematics Department, Norman, OK 73019-0315, USA |
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Abstract: | It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π2/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π2/3 due to the Gauss-Bonnet formula Vol(M) = 4π2/3 · χ(M). |
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Keywords: | Hyperbolic 4-manifolds volume function Poincaré's polyhedron theorem embedded totally geodesic hypersurfaces Gauss-Bonnet formula set of volumes |
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