4 1. The modular group and elliptic function theory
The case of infinity as a fixed point corresponds to a = oo and
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
is given in normal
form by
)z a
= K
A(z)~b '" z-b'
and equivalently as a matrix (of determinant 1)
A =
a - n2b ab{K2 - 1)
\ - K
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
As an example, we consider a sequence of nonparabolic motions
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
_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
K"n CVn\Kn ^n)
. - 1
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.
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