Parametrization of the Teichmüller Space of Bordered Surface NEC Groups

A non-Euclidean crystallographic group F(NEC group,for short)is a discrete subgroupof isometries of the hyperbolic plane■,with compact quotient space■.These groups uniformizeKlein surfaces,surfaces endowed with dianalytic structure.These surfaces can be seen as a generaliza-tion of Riemann surfaces.Fundamental polygons play an important role in the study of parametrizations of the Teichmiillerspace of NEC groups.In this work we construct a class of right-angled polygons which are fundamental regions of borderedsurface NEC groups.The free parameters used in the construction of the polygons give a parametriza-tion of the Teichmiiller space.From the parameters we obtain explicit matrices of the generators ofthe groups.Finally,we give examples to exhibit how different relations between the parameters reflectthe existence of automorphisms on the quotient surfaces.