Celebratio Mathematica

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.

W. D. Neumann and D. Zagier: “A note on an invariant of Fintushel and Stern,” pp. 241–244 in Geometry and topology (College Park, MD, 1983–1984). Edited by J. C. Alexander and J. L. Harer. Lecture Notes in Mathematics 1167. Springer, 1985. MR 827273 Zbl 0589.57016 incollection

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Low-dimensional topology and number theory (Oberwolfach, Germany, 15–21 August 2012), published as Oberwolfach Rep. 7 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2010. MR 3156734 Zbl 1209.00056 book

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Low-dimensional topology and number theory (Oberwolfach, Germany, 17–23 August 2014), published as Oberwolfach Rep. 11 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2014. MR 3444214 Zbl 1349.00178 book

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Low-dimensional topology and number theory (Oberwolfach, Germany, 20–26 August 2017), published as Oberwolfach Rep. 14 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2017. MR 3826628 Zbl 1394.00015 book

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Walter D. Neumann

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