Geodesic
Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere
Serdica Mathematical Journal, Tome 23 (1997) no. 2, pp. 143-164

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We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.
Keywords: Algebraic Curves, Abelian Threefolds, Period Matrices, Moduli Spaces, Shimura Surface, Siegel Domain, Complex Unit Ball, Uniformization, Braid Group, Monodromy Group, Modular Group, Gundamental Groups, Picard-Fuchsian Groups, Symplectic Group, Aritmetic Group, Representation, Quadratic Number Field 
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Rolf-Peter, Holzapfel. Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere. Serdica Mathematical Journal, Tome 23 (1997) no. 2, pp. 143-164. http://geodesic.mathdoc.fr/item/SMJ2_1997_23_2_a5/