A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endo...A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces. Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups. In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.展开更多
We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they ...We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.展开更多
文摘A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces. Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups. In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.
基金The first and third authors are partially supported by Project MTM2005-01637the second is partially supported by Projects Fondecyt 1030252,1030373 UTFSM 12.05.21
文摘We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.
