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 , … , \{ {P_k}\} ,k = 1,2,3, \ldots , and associate to each P k {P_k} an “integer” ν k {\nu _k} (which may be 1 , 2 , 3 , … , or ∞ ) 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 ν k {\nu _k} . More precisely, keeping the points P k {P_k} fixed, we vary the numbers ν k {\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.