4 1. The modular group and elliptic function theory

The case of infinity as a fixed point corresponds to a = oo and

—a2

a finite

and nonzero (this latter quantity is the translation length in this case); thus

a = 0 = aa. We record for future use

A'(a) = 1 and A"(a) = -2a.

A motion A with fixed points a and b and multiplier

K2

is given in normal

form by

A(z)

)z — a

= K

A(z)~b '" z-b'

and equivalently as a matrix (of determinant 1)

A =

a - n2b ab{K2 - 1)

\ - K

1

K

a

n{a — b)

Simple calculations show that

A'{a) = K2 and A\b) = *T 2 .

It follows that, unless A has order two, the selection of a multiplier allows

us to distinguish one fixed point from the other. For loxodromic A, we call

a the attractive (respectively, repulsive) fixed point provided |-A7(a)| 1

(respectively,

|-A;(a)|

1).

As an example, we consider a sequence of nonparabolic motions

A-n. —

an on

e PSL(2,

whose elements do not fix oo and that converges to a parabolic motion with

fixed point a. This means that (we may assume after multiplying the entries

of An by —1, if necessary)

j j

m

_H_ 1 —

a a n c

i Ji

m an

_|_ dn — 2.

n—oo 2cn n^oo

If, on the other hand, An fixes oo and an and a = oo, then

An —

K"n CVn\Kn ^n)

0

. - 1

with

lim Kn — 1, lim an = oo and lim an(Knl — nn) — b G C*

2. Riemann surfaces

More on the material in this section can be found in [6, Ch. I].

Definition 2.1. A Riemann surface M is a one (complex) dimensional con-

nected complex analytic manifold.