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The content of this monograph is situated in the intersection of
important branches of mathematics like the theory of one complex
variable, algebraic geometry, low dimensional topology and, from
the point of view of the techniques used, com- natorial group
theory. The main tool comes from the Uniformization Theorem for
Riemannsurfaces,
whichrelatesthetopologyofRiemannsurfacesandholomorphic or
antiholomorphic actions on them to the algebra of classical
cocompact Fuchsian groups or, more generally, non-euclidean
crystallographic groups. Foundations of this relationship were
established by A. M. Macbeath in the early sixties and dev- oped
later by, among others, D. Singerman. Another important result in
Riemann surface theory is the connection between
Riemannsurfacesandtheir symmetrieswith
complexalgebraiccurvesandtheirreal forms. Namely, there is a well
known functorial bijective correspondence between compact Riemann
surfaces and smooth, irreducible complex projective curves. The
fact that a Riemann surface has a symmetry means, under this
equivalence, that the corresponding complex algebraic curve has a
real form, that is, it is the complex- cation of a real algebraic
curve. Moreover, symmetries which are non-conjugate in the full
group of automorphisms of the Riemann surface, correspond to real
forms which are birationally non-isomorphic over the reals.
Furthermore, the set of points
xedbyasymmetryishomeomorphictoaprojectivesmoothmodeloftherealform
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