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action of D(x algebraic reflection analytic manifold apply Lemma applying H(U Bl B2 boundary curves iff canonical homology basis Chapter Cornell University Corollary 3.10 cp(D+(X cp(g curves iff cp(f desired result easy to verify element of Sn entries equivalence class equivalence relation exists h e D+(X extended homogeneous symplectic f e D(X fixed point set GL(n group of diffeomorphisms handles and r+1 holomorphic 1-forms homogeneous symplectic group hypothesis and Lemma identify each point implies induced action induction hypothesis Jordan curve let D+(X lower right hand non-orientable surface non-singular normal subgroup numbers orientable surface orientation reversing period matrices Proof of Theorem prove pullback quadratic form quasi-reflection QUASI-SYMMETRIC RIEMANN SURFACES quotient space r+1 boundary curves reflection resp Robert Zarrow simple calculation shows skew-symmetric Sn resp surface of genus SYMMETRIC AND QUASI-SYMMETRIC symplectic modular group Theorem 2.1 thesis Tn is equivalent tTZ+tR)(tSZ+tQ)"1 unimodular