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Arbitrarily Long Factorizations in Mapping Class Groups

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2015-01-01
On a compact oriented surface of genus g with n= 1 boundary components, d1, d2,..., dn, we consider positive factorizations of the boundary multitwist td1 td2 tdn, where tdi is the positive Dehn twist about the boundary di. We prove that for g= 3, the boundary multitwist td1 td2 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.