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In this paper we consider the topological class, denoted by AV2, of universalcovering maps from the plane to the sphere with two removed points. We provethat an element f in AV2 with finite post-singular set is combinatoriallyequivalent to a meromorphic transcendental map g with constant Schwarzianderivative if any only if f satisfies a Bounded Geometry condition.

Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values