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2017 The unstabilized canonical Heegaard splitting of a mapping torus
Yanqing Zou
Algebr. Geom. Topol. 17(6): 3435-3448 (2017). DOI: 10.2140/agt.2017.17.3435

Abstract

Let S be a closed orientable surface of genus at least 2. The action of an automorphism f on the curve complex of S is an isometry. Via this isometric action on the curve complex, a translation length is defined on f. The geometry of the mapping torus Mf depends on f. As it turns out, the structure of the minimal-genus Heegaard splitting also depends on f: the canonical Heegaard splitting of Mf, constructed from two parallel copies of S, is sometimes stabilized and sometimes unstabilized. We give an example of an infinite family of automorphisms for which the canonical Heegaard splitting of the mapping torus is stabilized. Interestingly, complexity bounds on f provide insight into the stability of the canonical Heegaard splitting of Mf. Using combinatorial techniques developed on 3–manifolds, we prove that if the translation length of f is at least 8, then the canonical Heegaard splitting of Mf is unstabilized.

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Yanqing Zou. "The unstabilized canonical Heegaard splitting of a mapping torus." Algebr. Geom. Topol. 17 (6) 3435 - 3448, 2017. https://doi.org/10.2140/agt.2017.17.3435

Information

Received: 23 April 2016; Revised: 14 May 2017; Accepted: 25 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1383.57021
MathSciNet: MR3709651
Digital Object Identifier: 10.2140/agt.2017.17.3435

Subjects:
Primary: 57M27
Secondary: 57M50

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 6 • 2017
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