Effective generation of right-angled artin groups in mapping class groups

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We show that given a collection \(X=\{f_1\), ..., \(f_m\}\) of pure mapping classes on a surface S, there is an explicit constant N, depending only on X, such that their Nth powers \(\{f_1^N\), ..., \(f_m^N\}\) generate the expected right-angled Artin subgroup of MCG(S). Moreover, we show that these subgroups are undistorted, and that each element is pseudo-Anosov on the largest possible subsurface.