Linearly Bounded Conjugator Property for Mapping Class Groups

Given two conjugate mapping classes f and g, we produce a conjugating element ω such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\omega| \le K (|f| + |g|)}$$\end{document}, where | · | denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.