| Let S be a connected and oriented surface of finite type. The mapping class group MCG&parl0;S&parr0; of S is the group of orientation-preserving homeomorphisms of S modulo isotopy. We study the coarse geometry of MCG&parl0;S&parr0; and obtain a uniform linear bound on the length of a conjugating element. This result establishes another positive analogy between MCG&parl0;S&parr0; and word hyperbolic groups, and has the important consequence that the conjugacy problem for MCG&parl0;S&parr0; is exponentially bounded. The main technical part of our theorem involves the torsion elements of MCG&parl0;S&parr0; . Our approach is to study the action of MCG&parl0;S&parr0; on various complexes associated to S . Connecting the geometry of MCG&parl0;S&parr0; as a metric space to the geometry of these complexes is the machinery of hierarchies. We give a careful analysis of hierarchies and establish some geometric properties for elements of MCG&parl0;S&parr0; . |
