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By Kolberg Michael

On_the_Moduli_Space_of_Cyclic_Trigonal_Riemann_Surfaces_of_Genus_4.(ISBN 9185643386)

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M′r,sr ) if and only if there exists a transitive permutation representation θ : Γ → ΣN satisfying the following conditions: 1. The permutation θ(xi ) has precisely si cycles of lengths less than mi , the lengths of these cycles being mi /m′i,1 , . . , mi /m′i,si . 2. The Riemann-Hurwitz formula µ(Γ′ ) =N µ(Γ) is satisfied. 5 the quotient space of H with a Fuchsian group Γ is a Riemann surface and the natural projection π : H → H/Γ is a holomorphic map. The importance of Fuchsian surface groups is made apparent from the following theorem which is a start of our classification of Riemann surfaces.

14. Let Γ ⊂ P SL2 (R) be an arbitrary Fuchsian group. 1. The Teichm¨ uller space of Γ, T (Γ) (Γ non-elementary), is L∞ (H, Γ)1 / ∼, where µ ∼ ν if and only if µ|R = ν|R (That is if Eµ = Eν ). T (Γ) is given the quotient topology from the Banach-norm topology on L∞ (H, Γ)1 . The quotient pseudometric τ on T (Γ) obtained from τˆ is again a metric and τ also induces the quotient topology from L∞ (H, Γ)1 . 16 The same being biholomorphically equivalent in this context. November 8, 2006 (9:45) 49 50 CHAPTER 1.

Similarly for the modular group of a Fuchsian group Γ we can state that M od(Γ) = Aut Γ/Inn(Γ) where Inn(Γ) is the normal subgroup of Aut Γ consisting of all inner automorphisms21 of Γ. Action of the modular group Clearly, T (X) does not depend on the reference Riemann surface X, but only on the quasiconformal type of X. Thus it makes sense of talking of [X, 1, X] ∈ T (X) as a base point for T (X). Now if ϕ : X → Y is a quasiconformal homeomorphism, there is a natural map ϕ∗ : T (X) → T (Y ) given by ϕ∗ ([X, f, X1 ]) = [Y, f ◦ ϕ, X1 ].