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Discontinuous Groups
An analytic function f is called automorphic with respect to a group Î of transformations of the plane if f takes the same value at points that are equivalent under Î. That is,
for each V â Î and each z â D, the domain of f. If we want to have nonconstant functions f, we must assume there are only finitely many equivalents of z lying in any compact part of D. This property of Î is known as discontinuity.
The most important domain for f from the standpoint of applications is the upper half-plane â H. Now from (1) with f analytic in H we deduce that V is analytic in H and maps H into itself. It is natural to require that V be one-to-one in order that Vâ1 should be single-valued. Hence F is a linear-fractional transformation. The group Î will therefore be a group of linear-fractional transformations, or as we shall call them, linear transformations.
The present chapter is devoted to a study of discontinuous groups of linear transformations.
1. Linear Transformations
1A. A linear transformation is a nonconstant rational function of degree 1; that is, a function
where Îą, β, Îł, δ are complex numbers and z is a complex variable. The function w is defined on all of the complex sphere Z except z = âδ/Îł and z = â. With the usual convention that w/0 = â for u â 0, we have
and we obtain w(â) by continuity:
In particular, âδ/Îł will be â if and only if Îł = 0, and in that case Îą/Îł = â: the infinite points of the two planes then correspond under the mapping w.
As a rational function, w is regular in Z except for a simple pole at z = âδ/Îł. Suppose Îł = 0. Then necessarily δ â 0, Îą â 0 (because of ιδ â βγ â 0) and
Hence dw/dz = Îą/δ â 0 and w is conformal at every finite z. At infinity we must use the uniformizing variables zⲠ= l/z, wⲠ= 1/w; then we find that
and wⲠis conformal at zⲠ= 0, which by definition means that w is conformal at z = â.
If Îł â 0, we have
hence dw/dz â 0 and w is conformal except possibly at z = âδ/Îł, z = â. At z = â δ/Îł we must use the variables zⲠ= z + δ/Îł, wⲠ= 1/w:
so that (dwâ˛/dzâ˛)z = 0 â 0. At z = â the correct variables are zⲠ= 1/z, wⲠ= w and we get
yielding the same conclusion.
Solving (2) for z we get
which is also a linear transformation and so is defined on the extended w-plane. The mapping z â w is therefore onto and hence one-to-one, and we may write z = wâ1.
Putting these results together we can assert:
THEOREM. The linear transformation (2) is a one-to-one conformal mapping of all of Zon itself.
For this reason a linear transformation is also called a con...