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CHAPTER 1. INTRODUCTION

1.2 Overview of the intertwining cases

The treatment assumes a hierarchy of cases. That is, case I is viewed as

being easier than case II, case II easier than case III, etc. If one begins

with a pair of generators in any of cases I-VI, one of three things can

happen: (i) one can tell immediately that the group is discrete by

using the Poincare polygon theorem, (ii) one can tell immediately that

the group is not discrete by using J0rgensen's inequality, the Shimizu-

Leutbecher theorem (p. 19 of [26]), or an area inequality, or (iii) one

cannot determine either fact directly from the given set of generators.

In the latter case one replaces the given set of generators (A, B) (after

some normalization) by the Nielsen equivalent pair (A, AB). Either

the new pair lies in an easier case or it remains in the same case. One

proves that one can only remain in the same case a finite number of

times before one is either forced to move up into an easier case or has

a situation of type (i) or (ii).

While pairs of elliptics and pairs of parabolics can be given treat-

ments that do not require moving up into an easier case, to give the

full treatment of case (VI) one needs to move through all of the cases

(I)-(VI), whence the name intertwining cases. Thus (VII) is the only

fully self-contained case.

In all seven cases one begins by enlarging the group G to a three

generator group, G*, in which G sits as a subgroup of index at most

two. G and G* are either simultaneously discrete or simultaneously

non-discrete. The construction of G* varies from case to case. When A

and B are pairs of hyperbolics with disjoint axes, for example, let L be

the common perpendicular to the two axes and let r^ denote reflection

in L. Then there are reflections r^ and r# such that A = r^ • r^ and

B = ri • r# and G* is the group generated by the three reflections.

The three possibilities (i), (ii), and (iii) usually correspond to three

different algebraic inequalities and geometric configurations. We illus-

trate this by considering the case of two hyperbolic generators A and

B whose axes are disjoint and whose product is hyperbolic. Since this

case has been treated erroneously in the literature, the example also

serves to illustrate how a point that is subtle from the algebraic point

of view is easily understood from the geometric perspective.

We let g and h be elements of 51/(2, R) whose images are A and