Poincaré's theorem for the modular group of real Riemann surfaces

Abstract

Let $Mo{d}_g$ be the modular group of surfaces of genus $g$. Each element $[h]\in Mo{d}_g$ induces in the integer homology of a surface of genus $g$ a symplectic automorphism $H([h])$ and Poincaré shown that $H:Mo{d}_g\to Sp(2g,\mathbb{Z})$ is an epimorphism. The theory of real algebraic curves justify the definition of real Riemann surface as a Riemann surface $S$ with an anticonformal involution $\sigma$. Let $(S,\sigma )$ be a real Riemann surface, the subgroup $Mo{d}_g^{\sigma }$ of $Mo{d}_g$ that consists of the elements $[h]\in Mo{d}_g$ that have a representant $h$ such that $h\circ \sigma =\sigma \circ h$, plays the rôle of the modular group in the theory of real Riemann surfaces. In this work we describe the image by $H$ of $Mo{d}_g^{\sigma }$. Such image depends on the topological type of the involution $\sigma$.

17 pages, LaTex