Abstract
We describe a method for constructing an arbitrary number of closed hyperbolic 3-manifolds of the same volume. In fact we prove that many hyperbolic 3-manifolds of finite volume have an arbitrary number of non-homeomorphic finite convering spaces of the same degree and hence the same volume. This applies, for example, to all hyperbolic 3-manifolds whose universal covering group is a subgroup of finite index in a Coxeter group generated by the reflections in the faces of a hyperbolic Coxeter polyhedron. It also applies to all hyperbolic 3-manifolds of finite volume with at least one cusp.
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Zimmermann, B. A note on hyperbolic 3-manifolds of the same volume. Monatshefte für Mathematik 117, 139–143 (1994). https://doi.org/10.1007/BF01299317
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DOI: https://doi.org/10.1007/BF01299317