By Flaass D.G.

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**Example text**

Set ω(x) = τ (x)−τ (x1 ) m τ (x)−τ (x2 ) m τ (x)−τ (x1 ) 2 τ (x)−τ (x2 ) η(x) , x2 is not a ramification point η(x) , x2 is a ramification point The proof that ϕX is immersive goes in the same way. 19 (i) Unravelling the definitions, the injectivity of ϕX is equivalent to the statement that for all x1 , x2 ∈ X and coordinates z1 , z2 in neighbourhoods of x1 , x2 , there exists a constant ω ω λ such that dz1 (x1 ) = λ dz2 (x2 ) for all ω ∈ Ω(X) if and only if τ (x1 ) = τ (x2 ) . Immersiveness, is equivalent to the statement that for all x ∈ X , there is an ω ∈ Ω(X) with a simple zero at x if and only if ı(x) = x .

Let tj ∈ (0, 1) , j ≥ 1 . Suppose that for all j ≥ 1 there exists an injective , holomorphic map φj : H(tj ) −→ X such that tj e−iθ tj eiθ , φj 0 ≤ θ ≤ 2π is homologous to Aj and φk H(tk ) ∩ φ H(t ) = ∅ for all k = . Then, n , Im RX n 1 2π ≥ j≥1 | log tj | n2j for all vectors n ∈ ZZ∞ with only a finite number of nonzero components. Proof: Let n ∈ ZZ∞ with only a finite number of nonzero components.

If the preimage is unbounded, there is a subsequence converging to an ideal boundary point p . 23, a = wp . 25 Let X be a parabolic Riemann surface with finite ideal boundary such that no ideal boundary point can be represented by a planar end. Let f be a meromorphic function on X with either two simple poles or one double pole and no other singularities. If f is bounded on the complement of any neighborhood of its poles, then X is hyperelliptic. Proof: Let I be the set of limits of f at the ideal boundary points of X .

### 2-Local subgroups of Fischer groups by Flaass D.G.

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