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
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