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Spectral geometry, link complements and surgery diagrams. (English) Zbl 1211.57011

The paper under review finds bounds on the first eigenvalue of the Laplacian and the Cheeger constant for finite volume hyperbolic 3-manifolds. The construction of expanding families of manifolds (infinite families with first eigenvalue of the Laplacian bounded away from zero) is a major research topic that involves many fields, as group theory, number theory and coding theory. In the case of hyperbolic manifolds, the bounds of the Cheeger constant are equivalent to those of the first eigenvalue of the Laplacian. Previous work of the author connects this topic for hyperbolic three manifolds with the virtual Haken conjecture. The first bound of the Cheeger constant for a hyperbolic 3-manifold \(M\) obtained in the paper is in terms of the volume of \(M\) and the combinatorial data of the link diagram of \(M\). In particular, this applies to links in \(S^3\).
This is further studied for two families of links, those with an alternating diagram, and those with a highly twisted diagram, where the volume of the link exterior is also bounded in terms of combinatorial data of a link diagram, by M. Lackenby [Proc. Lond. Math. Soc., III. Ser. 88, No. 1, 204–224 (2004; Zbl 1041.57002)]. For those links, the bounds of the Cheeger constant or of the first eigenvalue of the Laplacian are obtained in terms of combinatorial data of the diagrams. Those results are applied to the study of expanding families. As a first application, a family of hyperbolic alternating (or highly twisted) link exteriors is expanding if and only if their volumes are bounded above. Thus the author rises the question whether there exist families of hyperbolic link complements with unbounded volume and forming an expanding family. Another application is the following finiteness result: there are only finitely many alternating or highly twisted link complements that are congruence arithmetic hyperbolic three manifolds.
It is also proved that a recent result of F. Costantino and D. Thurston [J. Topol. 1, No. 3, 703–745 (2008; Zbl 1166.57016)] is sharp. Namely, Theorem 1.10: Let \(\{M_i\}\) be an expanding family of finite-volume orientable hyperbolic manifolds. Then, there is a positive constant \(c\) such that any rational surgery diagram for \(M_i\) requires at least \(c(volume(M_i))^2\) crossings. (Costantino and Thurston proved that the growth is at most quadratic).
The proof is based in the relationship between different notions of width: for links in the three sphere, for abstract graphs, for planar graphs and for Heegaard splittings. The separator theorem for planar graphs of R. J. Lipton and R. E. Tarjan [SIAM J. Appl. Math. 36, 177–189 (1979; Zbl 0432.05022)] is also used. The paper is very clear and well written.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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