Piecewise-euclidean structures on Riemann surfaces

It is well-known that a Riemann metric defines the complex structure on a smooth surface; similarly, one can define the complex structure on a PL-surface by a piecewise-euclidean metric. In the talk this construction will be presented and it will be demonstrated that all the complex structures can be thus obtained. According to Riemann's existence theorem this construction associates the complex algebraic curve to any piecewise-euclidean structure. In the talk the special piecewise-euclidean structures will be considered, namely, defined by the equilateral triangulations and by the square quadrangulations. It will be shown that the corresponding algebraic curves are the complexifications of curves defined over the field of algebraic numbers and, conversely, that any curve over the field of algebraic numbers can be defined by an equilateral as well as by a square structure. Some examples will be presented.