Minimum ideal triangulations of hyperbolic 3-manifolds
Let \(\sigma\) (n), resp. \(\sigma_{or}(n)\) denote the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volume n- cusped hyperbolic 3-manifold, resp. such an orientable one. The authors show that \(2n-1\leq \sigma (n)\leq \sigma_{or}(n)\leq 4n-4\) for \(n\geq 5\), and that \(\sigma_{or}(n)\geq 2n\) for all n. For \(n=1,2,3,4,5\) they give correct values of \(\sigma\) (n) and \(\sigma_{or}(n)\) and the corresponding manifolds in tables. \(\sigma (1)=1\), \(\sigma (2)=2\), \(\sigma (3)=4\), \(\sigma (4)=6\) have unique nonorientable realizations. \(\sigma (5)=\sigma_{or}(5)=10\) corresponds to one orientable manifold (tabulated) and at least two non-orientable ones. \(\sigma_{or}(1)=2\) and \(\sigma_{or}(2)=4\) each has two realizations. \(\sigma_{or}(3)=6\) and \(\sigma_{or}(4)=8\) appear with examples by link complements. The proof of minimality is simple combinatorial, except \(\sigma (5)=10\) which is not detailed here. These investigations are in connection with those of Thurston's circle. The reviewer lacks a reference to the paper of \textit{S. V. Matveev} and \textit{A. T. Fomenko} [Russ. Math. Surv. 43, No.1, 3-24 (1988); translation from Usp. Mat. Nauk. 43, No.1, 5-22 (1988; Zbl 0671.58008)].
scientific article; zbMATH DE number 4190576
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