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, combinatorial
group theory. The main tool comes from the Uniformization Theorem for
Riemann surfaces, which relates the topology of Riemann surfaces and holomorphic
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 developed
later by, among others, D. Singerman.