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Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds. (English) Zbl 1446.57012

Summary: T. H. Colding and D. Gabai [Duke Math. J. 167, No. 15, 2793–2832 (2018; Zbl 1403.57012)] have given an effective version of T. Li’s theorem [J. Am. Math. Soc. 19, No. 3, 625–657 (2006; Zbl 1108.57015)] that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces \(S_i\) and incompressible surfaces \(K_j\) such that any strongly irreducible Heegaard surface is a Haken sum \(S_i+\sum_jn_jK_j\), up to one-sided associates of the Heegaard surfaces.

MSC:

57K30 General topology of 3-manifolds
57K32 Hyperbolic 3-manifolds
57M50 General geometric structures on low-dimensional manifolds
57M99 General low-dimensional topology
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References:

[1] Colding, Tobias Holck; Gabai, David, Effective finiteness of irreducible Heegaard splittings of non-Haken 3-manifolds, Duke Math. J., 167, 15, 2793-2832 (2018) · Zbl 1403.57012 · doi:10.1215/00127094-2018-0022
[2] Li, Tao, Heegaard surfaces and measured laminations. II. Non-Haken 3-manifolds, J. Amer. Math. Soc., 19, 3, 625-657 (2006) · Zbl 1108.57015 · doi:10.1090/S0894-0347-06-00520-0
[3] Moriah, Yoav; Schleimer, Saul; Sedgwick, Eric, Heegaard splittings of the form \(H+nK\), Comm. Anal. Geom., 14, 2, 215-247 (2006) · Zbl 1119.57008
[4] Oertel, U., Incompressible branched surfaces, Invent. Math., 76, 3, 385-410 (1984) · Zbl 0539.57006 · doi:10.1007/BF01388466
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