## 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.

### MSC:

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

### Citations:

Zbl 0910.57006; Zbl 1037.57015
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