The Coarse Geometry of the Teichmuller Metric: A Quasiisometry Model and the Actions of Finite Groups

Let S be a surface of finite type and T(S) its Teichmuller space. In the first chapter of the thesis, we build a graph called the augmented marking complex which is quasiisometric to Teichmuller space with the Teichmuller metric. In the second chapter, we analyze the sublevel sets of the diameter map of the action of a finite order subgroup of the mapping class group. Our main theorem in this chapter proves that each sublevel set lives in a bounded diameter neighborhood of the fixed point set, where the bound depends only on the sublevel constant and the surface.