Se p 20 13 Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values ∗

In this paper we study the topological class of universal covering maps from the plane to the sphere with two removed points; we call the elements topological transcendental maps with two asymptotic values and we denote the space byAV2. We prove that an element f ∈ AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if and only if f satisfies an analytic condition we call bounded geometry. We plan to relate the bounded geometry condition to topological conditions such as Levy cycles and Thurston obstructions and to the geometric condition called a canonical Thurston obstruction in a future paper.