Poincare’s theorem for the modular group of real Riemann surfacesстатья
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Аннотация:Let Mod(g) denote the modular group of (closed and orientable) surfaces S of genus g. Each element vertical bar h vertical bar is an element of Mod(g) induces a symplectic automorphism H(vertical bar h vertical bar) of H(1)(S, Z). Poincare showed that H : Mod(g) -> Sp(2g, Z) is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution sigma. Let (S, sigma) be a real Riemann surface, Homeo(g)(sigma) be the group of orientation preserving homeomorphisms of S such that h o sigma = sigma o h and Homeo(g.0)(sigma) be the subgroup of Homeo(g)(sigma) consisting of those isotopic to the identity by an isotopy in Homeo(g)(sigma). The group Mod(g)(sigma) = Homeo(g)(sigma) / Homeo(g.0)(sigma) plays the role of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H of Mod(g)(sigma). Such image depends on the topological type of the involution sigma. (C) 2009 Elsevier B.V. All rights reserved.