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Large embedded balls and Heegaard genus in negative curvature. (English) Zbl 1056.57014

Let \(M\) be a closed connected orientable 3-manifold which admits a hyperbolic structure. By Mostow rigidity the hyperbolic metric on \(M\) is unique up to isometry. Thus geometric invariants of the hyperbolic metric are actually topological invariants of \(M\). One may attempt to relate geometric invariants to other topological invariants. For example in D. Cooper [Proc. Am. Math. Soc. 127, 941–942 (1999; Zbl 1058.30037)], it is shown that the volume of \(M\) is at most \(\pi\) times the length \(L\) of any presentation of the fundamental group of \(M\). A result of M. Lackenby [Proc. Lond. Math. Soc., III. Ser. 88, No. 1, 204–224 (2004; Zbl 1041.57002)] shows that for alternating links the volume is bounded above and below by explicit affine functions of a certain combinatorial invariant: the twist number. The injectivity radius of \(M\) is defined to be the radius of the smallest self-tangent isometrically embedded ball in \(M\). In M. E. White [Commun. Anal. Geom. 10, No. 2, 377–395 (2002; Zbl 1012.57020)] it is shown that the injectivity radius of \(M\) is bounded above by a function of the rank of its fundamental group (which is the minimum number of generators required to generate the group). Since the Heegaard genus is always at least as large as the rank, White’s result gives a corresponding upper bound on injectivity radius in terms of Heegaard genus.
In this paper the authors show that if \(M\) is a closed connected orientable hyperbolic 3-manifold with Heegaard genus \(g\), then \(g\geq {1\over 2}\cosh(r)\) where \(r\) denotes the radius of any isometrically embedded ball in \(M\). Assuming an unpublished result of Pitts and Rubinstein improves this to \(g\geq{\cosh(r)+ 1\over 2}\). Finally, the authors give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N16 Geometric structures on manifolds of high or arbitrary dimension
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References:

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