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


An orientation reversing involution  s  of a topological compact genus g,  g>2, surface  S  induces an antiholomorphic involution

of the Teichmüller space of of genus g Riemann surfaces. Two such involutions  s* and t* are conjugate in the mapping class group if and  only  if  the corresponding orientation reversing involutions    s  and  t  of   S are conjugate in the automorphism group of  S. This is equivalent to saying that the quotient surfaces

are homeomorphic.   Hence the   Tg  has  mg = [2 + 3g/2]  distinct antiholomorphic involutions,  which are also called real structures of Tg.

This result is a simple fact that follows from  Royden's theorem stating that the the mapping class group is the full group of  holomorphic automorphisms  of the  Teichmüller space (g>2).


be two real structures  that are not conjugate in the mapping class group.  In this paper  we construct   a real analytic diffeomorphism d: Tg  -> Tg such that


This mapping   d   is a product of  full and half Dehn-twists around certain simple closed curves on the surface   S.

This  result has applications to the moduli spaces of real algebraic curves.