1.1. OVERVIEW INTERSECTING AXES 5

discreteness problem requires an algorithmic solution. This requires

an elementary introduction to the theory of algorithms and models of

computation which is found in the introductary sections of chapter 14

(through sections 14.1 and 14.1.2) along with section 1.3 of this chapter.

Since this monograph has four main goals, readers may consult this

manuscript for different reasons. A reader who has a two generator sub-

group of P5X(2,R ) and wants to know whether or not it is discrete,

will only need to look at the real number algorithm or the Turing ma-

chine algorithm (theorem 14.4.1 or theorem 14.5.1 of Part IV) or the

discreteness theorem (theorem 3.1.1). A reader who wants an overview

of the procedure and/or an understanding of the relationship between

the algebraic solution and the more geometrically motivated approach

will read all of Part I which consists of the introduction (chapter 1), the

proof that the triangle algorithm stops (chapter 2) and the outline of

the proof of the discreteness theorem (chapter 3). Others may shorten

this to the overviews in this introduction, especially the overview of the

intertwining case (section 1.2), and the statements of the main theo-

rems in chapters 2 and 3. Still others who want to see the complete

proof of the discreteness theorem for hyperbolics with intersecting axes

will read all of Parts I, II and III using chapter 3 of Part I as a guide to

the contents of Parts II and III. Finally those who want a better under-

standing of the role that algorithms play will read part IV, especially

the introductory sections of chapter 14.

An attempt has been made to organize the material so as to make

it accessible to all types of readers. This has required some repetition

to assure that certain sections could be read independently. It is hoped

that this repetition has been kept to a minimum.

1.1 Overview of hyperbolics with inter-

secting axes

The discreteness or non-discreteness of the group generated by a pair

of hyperbolics with intersecting but distinct axes depends upon the

nature of the commutator of the two generators. If one begins with

a pair of hyperbolics with intersecting axes, one of three things can