The unstabilized canonical Heegaard splitting of a mapping torus

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.