Linearly Bounded Conjugator Property for Mapping Class Groups

Given two conjugate mapping classes f and g, we produce a conjugating element ω such that ${|\omega| \le K (|f| + |g|)}$ , 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.