Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

Abstract

An orientation reversing involution \(\sigma\) of a topological compact genus \(g,\, g>2,\) surface \(\Sigma\) induces an antiholomorphic involution \(\sigma^*: T^g \to T^g\) of the Teichmuller space of genus g Riemann surfaces. Two such involutions \(\sigma^*\) and \(\tau^*\) are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions \(\sigma\) and \(\tau\) of \(\Sigma\) are conjugate in the automorphism group of \(\Sigma\). This is equivalent to saying that the quotient surfaces \(\Sigma/\langle \sigma\rangle\) and \(\Sigma/\langle \tau\rangle\) are homeomorphic. Hence the Teichmuller space \(T^g\) has \(m_g = \lfloor{3g+4\over 2}\rfloor\) distinct antiholomorphic involutions, which are also called real structures of \(T^g\) ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmuller space (\(g>2\)). Let \(\sigma^*: T^g\to T^g\) and \(\tau^*: T^g\to T^g\) be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism \(d: T^g\to T^g\) such that