Surface bundles versus Heegaard splittings

Abstract

This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting. As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy, then the splitting is standard.

The rank of the fundamental group of certain hyperbolic 3–manifolds fibering over the circle

“…It should be said that in fact most arguments here are found in some form in the papers of Kapovich and Weidmann and that the main result of this note cannot come as a surprise to these authors. It should also be mentioned that a more general result in the spirit of Theorem 1.1, but in the setting of Heegaard splittings, is due to Bachmann and Schleimer [2].…”

Tunnel number one, genus-one fibered knots

“…Because this method has been described in detail elsewhere, we will give only an outline of the setup and leave many of the details to the reader. A similar exposition for general surface bundles can also be found in [1].…”

“…Nevertheless, observe that adding two full positive twists to β gives the braid (σ 2 σ 1 σ 2 ) 4 β −3,5 = β 1,5 . Remark 3.3 thus implies that K is the core of a (−1)-Dehn surgery on the lift of the braid axis ofβ 1,5 to the double branched cover, i.e. the boundary of the plumbing of a 7-Hopf band and a (+1)-Hopf band.…”

Fibered faces, veering triangulations, and the arc complex

“…Bachman-Schleimer [BS05] use Heegaard surfaces to give bounds on the curve-complex translation distance of the monodromy of a fibering. All of these bounds apply to entire surfaces and not to subsurface projections.…”