Arbitrarily Long Factorizations in Mapping Class Groups

On a compact oriented surface of genus $g$ with $n\geq 1$ boundary components, $\delta _1, \delta _2,\ldots , \delta _n$, 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 $g\geq 3$, 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 $g\geq 8$. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of contact three manifolds.