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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 T^{g}
has m_{g} = [2 + 3g/2] distinct antiholomorphic involutions,
which are also called real structures of T^{g}.
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).

Let

s^{*}: T^{g} -> T^{g} and
t ^{*}: T^{g} -> 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} -> T^{g} such that

s ^{* }= d ^{-1 } o
t^{*} o d.

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.