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3. UNIFORMIZATION θ θ : ∆ −→ G ↓ ↓ Γ −→ 1d The Fuchsian group ∆ is the lifting of G to the universal covering fu : H → H/Γ and is called the universal covering transformation group of (X, G). Hence we have that the orbit space X/G is uniformized by ∆, X/G = H/∆. H −→ 1 −→ X −→ X/G (H/Γ) (H/∆) Γ −→ ∆ The covering f : X → X/G = H/∆ is a regular branched covering with monodromy group G = ∆/Γ. 38. 4) above are named Hurwitz groups after the mathematician Adolf Hurwitz (1859-1919) who proved the bound in 1893.
It then makes sense talking about locally quasiconformal maps between such surfaces. In particular a conformal atlas is quasiconformal. It also makes sense to speak about globally quasiconformal homeomorphisms between Riemann surfaces. Given a function (Beltrami coefficient) µ in L∞ (D)1 13 , the equation wz¯ = µ(z)wz is called the Beltrami equation with coefficient µ. It is natural to call any quasiconformal map w satisfying the above equation a µ-conformal homeomorphism of D. The following theorem shows that solutions to the Beltrami equation always exist.
As we have seen, when the action is free then all the stabilizers are trivial. 1. 8 The norm of a matrix A ∈ P SL2 (R) is defined as A = (a2 + b2 + c2 + d2 )1/2 . in a sense suitable hyperbolic polygon 10 Some books refer to free actions as discontinuous actions and then calls the weaker action for properly discontinuous. 9A November 8, 2006 (9:45) 33 34 CHAPTER 1. PRELIMINARIES The structure of Fuchsian groups A Fuchsian group Γ has a presentation with generators x1 , . . , xr , a1 , b1 , . .