Abstract:

This thesis contributes to the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators of the Fuchsian groups say Gt, associated to a lparameter family of compact Riemann surfaces Xt, where Xt are the Riemann surfaces for the complex algebraic curves arising from a lparameter family of irreducible polynomials. The main idea of this thesis is to utilize explicit quasiconformal mappings between algebraic curves, calculate the Beltrami coefficients, and hence utilize the AhlforsBers variational formulae when applied to quasiconformal conjugates of Fuchsian groups. The direct practical implementation of the variational formulae that is determined in this thesis is quite feasible. It explains how certain classical Poincare theta series with respect to the initial Fuchsian group can be brought to bear on this problem of applying these variational formulae in a computer package. Although the compact Riemannsurfaces are dealt with and the torsionfree parabolicfree Fuchsian uniformizing group, in this, The Theory of Teichmuller spaces work exactly the same for Riemann surfaces of finite conformal type. It could allow distinguished points or punctures on the compact Riemann surfaces and correspondingly allow elliptic or parabolic elements in the Fuchsian groups under scrutiny, and obtains exactly parallel results. One can directly apply the theorems developed in Chapter IV, to the present case in order to obtain the actual generators of the deformed Fuchsian groups that represent the deformations of the Fermat curves. 