Celebratio Mathematica — Neumann

A celebration of Walter Neumann

Works connected to Don Bernard Zagier

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W. D. Neumann and D. Zagier: “Volumes of hyperbolic three-manifolds,” Topology 24 : 3 (1985), pp. 307–332. MR 815482 Zbl 0589.57015 article

By “hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold \( M \) of finite volume. By Mostow rigidity the volume of \( M \) is a topological invariant, indeed a homotopy invariant, of the manifold \( M \). There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [Borel 1981; Zagier 1986]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [1977]. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes.