A continuity theorem for Fuchsian groups

Abstract:

On a given Riemann surface, fix a discrete (finite or infinite) sequence of points $\{ {P_k}\} ,k = 1,2,3, \ldots ,$ and associate to each ${P_k}$ an “integer” ${\nu _k}$ (which may be $1,2,3, \ldots ,{\text {or}}\;\infty )$. This sequence of points and “integers” is called a “signature” on the Riemann surface. With only a few exceptions, a Riemann surface with signature can always be represented by a Fuchsian group. We investigate here the dependence of the group on the number ${\nu _k}$. More precisely, keeping the points ${P_k}$ fixed, we vary the numbers ${\nu _k}$ in such a way that the signature tends to a limit signature. We shall prove that the corresponding representing Fuchsian group converges to the Fuchsian group which corresponds to the limit signature.