On the number of hyperbolic 3-manifolds of a given volume (Q2867797)

In the present paper the authors compute the number \(N(v)\) of complete orientable hyperbolic \(3\)-manifolds of volume \(v\) for an infinite sequence of volumes. Jørgensen and Thurston showed that \(N(v)\) is finite for all given volumes \(v\), but so far the only known positive value of \(N(v)\) was for the lowest volume \(v_1\) realized only by the Weeks manifold [\textit{D. Gabai} et al., Comment. Math. Helv. 86, No. 1, 145--188 (2011; Zbl 1207.57023)], [\textit{P. Milley}, J. Topol. 2, No. 1, 181--192 (2009; Zbl 1165.57016)].NEWLINENEWLINEThe authors construct an infinite sequence of closed hyperbolic \(3\)-manifolds determined by their volumes, which provides an infinite sequence of distinct volumes \(\{x_i\}\) such that \(N(x_i)=1\). Such a family of manifolds is obtained by Dehn filling on the figure eight knot complement. The proof uses the asymptotic formula of \textit{W. D. Neumann} and \textit{D. Zagier} [Topology 24, No. 3, 307--332 (1985; Zbl 0589.57015)] which is an asymptotic expansion for volume change under Dehn filling in terms of a quadratic form associated to the Dehn filling. Studying this formula the authors show that there exists a family of Dehn fillings on the figure eight knot complement uniquely determined by their volumes \(x_i\) amongst Dehn fillings on the figure eight knot complement and its sister. By Thurston's result on volume of Dehn fillings, \(x_i\) converges from below to the volume \(v_\omega\) of the figure eight knot complement and its sister. \textit{C. Cao} and \textit{G. R. Meyerhoff} [Invent. Math. 146, No. 3, 451--478 (2001; Zbl 1028.57010)] showed that these are the only two orientable cusped hyperbolic \(3\)-manifolds of smallest volume. Now the conclusion follows by the fact due to Jorgensen and Thurston that any orientable closed hyperbolic \(3\)-manifold with volume less than \(v_\omega\) but sufficiently close to \(v_\omega\) comes from Dehn filling on the figure eight knot complement or its sister.NEWLINENEWLINEFurthermore the authors study \(N(v)\) restricted to two special classes of orientable hyperbolic \(3\)-manifolds: cusped manifolds and link complements. They exhibit an infinite sequence of \(1\)-cusped hyperbolic \(3\)-manifolds determined by their volumes amongst orientable cusped hyperbolic \(3\)-manifolds. Such a family is obtained by Dehn filling on one component of the \((-2,3,8)\)-pretzel link complement. The proof is similar to the previous result but it requires a more accurate study of the Neumann-Zagier asymptotic expansion.NEWLINENEWLINEFinally, they show that for \(n\geq 3\) there are at least \(2^n/(2n)\) different hyperbolic link complements of volume \(4n\) times the volume \(V_8\) of the regular ideal octahedron, therefore the number of hyperbolic link complements of volume \(4nV_8\) grows at least exponentially with \(n\) going to infinity.NEWLINENEWLINEFor the entire collection see [Zbl 1272.57002].