A continuity theorem for fuchsian groupsc)

On a given Riemann surface, fix a discrete (finite or infinite) sequence of points {Pk}, k = 1, 2, 3,…, and associate to each Pk an “integer” vk (which may be 1, 2, 3,…, or ∞). 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 vk. More precisely, keeping the points Pk fixed, we vary the numbers vk 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. © 1972 American Mathematical Society.