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On_the_Moduli_Space_of_Cyclic_Trigonal_Riemann_Surfaces_of_Genus_4.(ISBN 9185643386)

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On_the_Moduli_Space_of_Cyclic_Trigonal_Riemann_Surfaces_of_Genus_4. (ISBN 9185643386)

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This means that Γµ = wµ ◦ Γ ◦ wµ−1 is again a discrete subgroup of P SL2 (R) if and only if µ ∈ L∞ (H, Γ)1 . Thus, each such µ determines a monomorphic15 embedding Eµ of Γ into P SL2 (R): Eµ : Γ → Γµ given by Eµ (g) = gµ = wµ ◦ g ◦ wµ−1 . Since conjugation by quasiconformal maps preserves the type of M¨ obius transformation, we know that the image group Γµ is also a Fuchsian group of the same signature as Γ. 14 If Φ : (M, d) → T is a surjection from a metric space (M, d) onto a set T , then the quotient pseudometric (via Φ) on T is defined to be dΦ (t1 , t2 ) = inf{d(m1 , M2 ) : m1 ∈ Φ−1 (t1 ), m2 ∈ Φ−1 (t2 )}.

These points form a single orbit for the action of G on X. Since the xi ’s are all in the same orbit, they have conjugate stabilizer subgroups, and in particular, each stabilizer subgroup is of the same order, say r. Moreover, the number s of points in the orbit is the index of the stabilizer, and so is equal to |G|/r. This proves the following. 6. Let G be a finite group acting holomorphically and effectively on a compact Riemann surface X, with quotient map π : X → Y = X/G. Then for every branch point y ∈ Y there is an integer r ≥ 2 such that π −1 (y) consists of exactly |G|/r points of X, and at each of the pre-image points, π has multiplicity r.

November 8, 2006 (9:45) 29 30 CHAPTER 1. 9: The ramification of the pre-images We can see how small loops around the branch point is lifted up to mj paths which together form a loop around each uj . So the cycle structure representing a small loop around b becomes (m1 )(m2 ) . . (mk ), that is each cycle is of length mj . The Euler characteristic of a Riemann surface The complex structure of a Riemann surface provides a geometrical structure as a 2-orbifold. We can extend the definition of the Euler characteristic to Riemann surfaces with cone points.

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