Braid groups and Hodge theory (Q1945159)

A configuration of distinct points \((b_1,\ldots, b_n)\) in the complex plane gives rise to a compact Riemann surface which is a branched covering of \(\mathbb{P}^1\) whose equation is \(y^d=(x-b_1)\ldots (x-d_n)\). The author studies the unitary representations of the braid group and the geometric structures on moduli space that arise from such branched coverings, developing new connections between the braid group and arithmetic groups, ergodic theory, complex reflection groups, Teichmüller curves, moduli spaces of 1-forms, period maps, plane polgyons, hypergeometric functions and Thurston's work on shapes of polyhedra. The approach uses the classification of certain arithmetic subgroups of \(U(r,s)\) which envelop the image of the braid groups. The paper is amazingly rich for the connections made between several theories. The author also mentions related open problems in surface topology.