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The asymptotic behaviour of Heegaard genus. (English) Zbl 1063.57002

The results of this paper concern two of the author’s conjectures that involve the Heegaard genera of the finite covering spaces of a compact orientable \(3\)-manifold \(M\). They are stated in terms of \(\chi_-^h(M_i)\), the negative of the maximal Euler characteristic of a Heegaard splitting of a covering space \(M_i\) of \(M\), and \(\chi_-^{sh}(M_i)\), the negative of the maximal Euler characteristic of a strongly irreducible Heegaard splitting of \(M_i\). Putting \(d_i\) equal to the degree of the covering map \(M_i\to M\), the conjectures state that \(\inf\chi_-^h(M_i)/d_i = 0\) for a compact hyperbolic \(M\) if and only if \(M\) virtually fibers over the circle, and that \(\lim\inf\chi_-^{sh}(M_i)/d_i>0\) for any closed orientable hyperbolic \(M\). In previous work, the author has shown these and corresponding conjectures to hold for various kinds of \(M\) or when restricted to certain types of coverings. In this paper, the following is established for any collection \(\{M_i\to M\}\) of finite regular coverings of a closed orientable \(3\)-manifold \(M\) with a negatively curved Riemannian metric: (1) if \(\chi_-^h(M_i)/\sqrt{d_i}\to 0\), then the first Betti numbers \(b_1(M_i)\) are positive for all sufficiently large \(i\), (2) \(\chi_-^{sh}(M_i)/\sqrt{d_i}\) is bounded away from \(0\), and (3) if \(\chi_-^h(M_i)/\sqrt 4{d_i}\to 0\), then \(M_i\) fibers over the circle for all sufficiently large \(i\). The proofs combine a variety of results and ideas about \(3\)-manifolds and Riemannian geometry, including generalized Heegaard splittings as developed by M. Scharlemann and A. Thompson, minimal surfaces (notably a nice result of J. Pitts and J. H. Rubinstein about minimal surfaces in \(3\)-manifolds with “bumpy” Riemannian metrics), the Cheeger constant, and techniques of the author for analyzing coverings using these tools.

MSC:

57M10 Covering spaces and low-dimensional topology
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
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