Fibred and virtually fibred hyperbolic 3-manifolds in the censuses. (English) Zbl 1085.57012

More than \(15,000\) examples of hyperbolic \(3\)-manifolds have been catalogued: for the closed case nearly \(11,000\) appear in the Hodgson-Weeks census (HW) and almost \(5,000\) cusped manifolds are in the Callahan-Hildebrand-Weeks census (CHW) [P. J. Callahan, M. V. Hildebrand, J. R. Weeks, Math. Comp. 68, No. 225, 321–332 (1999; Zbl 0910.57006)]. These provide a testing ground for conjectures, in particular for W. Thurston’s conjecture that every closed hyperbolic \(3\)-manifolds is virtually Haken (has a finite covering manifold which contains a two-sided incompressible surface), and his even stronger conjecture that all closed hyperbolic \(3\)-manifolds are virtually fibered over the circle. N. M. Dunfield and W. P. Thurston [Geom. Topol. 7, 399–441 (2003; Zbl 1037.57015)] verified that every manifold in CHW satisfies the virtually Haken conjecture. In more than 87% of those cases, they actually found a finite covering that fibers over the circle.
In the paper under review, the author combines a variety of theoretical and computational ideas to obtain a great deal of additional information. First, all fibered \(3\)-manifolds in both censuses are determined, by resolving the remaining unknown \(169\) cases in HW and \(128\) cases in CHW. This leads to significant new examples of virtually fibered nonfibred \(3\)-manifolds. Also, the corank (the maximum rank of a free group quotient) of the fundamental group is determined for all manifolds in the census. In fact, it is \(1\) for every manifold in the cusped case, and for the closed case it is \(0\) when the first Betti number is \(0\), and \(1\) otherwise. All of the methodology depends only on the fundamental group. Considerable information is extracted from the Alexander polynomial, which the author computes for the unknown cases by utilizing a number of clever observations. Another useful device is the Bieri-Neumann-Strebel invariant, which provides ways to test whether the kernel of a homomorphism to the group of integers is finitely generated. In the case of two-generator one-relator groups, it leads to a remarkably simple algorithm which can be hand-checked. Included tables provide the following: the Alexander polynomials of the previously unresolved examples, the closed fibered census \(3\)-manifolds, the \(41\) closed nonfibered census \(3\)-manifolds with infinite first homology, and the nonfibered but virtually fibered census manifolds.


57M50 General geometric structures on low-dimensional manifolds
57M05 Fundamental group, presentations, free differential calculus
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