#### Volume 17, issue 6 (2017)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
The unstabilized canonical Heegaard splitting of a mapping torus

### Yanqing Zou

Algebraic & Geometric Topology 17 (2017) 3435–3448
##### 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 ${M}_{f}$ depends on $f$. As it turns out, the structure of the minimal-genus Heegaard splitting also depends on $f$: the canonical Heegaard splitting of ${M}_{f}$, 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  ${M}_{f}$. 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 ${M}_{f}$ is unstabilized.

##### Keywords
Heegaard splitting, stabilization, mapping torus, translation length
Primary: 57M27
Secondary: 57M50
##### Publication
Received: 23 April 2016
Revised: 14 May 2017
Accepted: 25 May 2017
Published: 4 October 2017
##### Authors
 Yanqing Zou Department of Mathematics Dalian Minzu University Dalian China