Arbitrarily Long Factorizations in Mapping Class Groups

Abstract

On a compact oriented surface of genus $g$ with $n\ge 1$ boundary components, ${\delta }_1,{\delta }_2,\dots ,{\delta }_n$, we consider positive factorizations of the boundary multitwist $t_{{\delta }_1}{t}_{{\delta }_2}\cdots t_{{\delta }_n}$, where $t_{{\delta }_i}$ is the positive Dehn twist about the boundary ${\delta }_i$. We prove that for $g\ge 3$, the boundary multitwist $t_{{\delta }_1}{t}_{{\delta }_2}$ can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn-Morris, who proved this result for $g\ge 8$. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of conctact three manifolds.

13 pages, 4 figures, a few typos are corrected

References in corpus (2)

• Low-dimensional homology groups of mapping class groups: a survey
• Stein fillings of planar open books