Arbitrarily long factorizations in mapping class groups

Abstract

Abstract. On a compact oriented surface of genus g with n ≥ 1 boundary components, δ1, δ2, . . . , δn, we consider positive factorizations of the boundary multitwist tδ1 tδ2 · · ·tδn , where tδi is the positive Dehn twist about the boundary δi. We prove that for g ≥ 3, the boundary multitwist tδ1 tδ2 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 ≥ 8. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of contact three manifolds