INTRODUCTION 3

readers who are very familiar with Procesi's work will see more or less instantly

why arithmetic in

H[7r],

where

7r

is a free group, is the same thing as universal

arithmetic with determinant one matrices in a 2 x 2 matrix ring. Other readers, as

did the authors of this work, will find it necesary to wade through a hundred pages

of technical formulas before understanding

H[7r].

We are only partly joking, but in

some sense our labors seem worthwhile. We develop interesting finite presentations

of

H[1r]

when

~ E

k, for all finitely presented groups

1r,

including some results on

H[7r]

for free groups

7r

which perhaps extend the known computational results from

invariant theory of 2 x 2 matrices.

We now resume our outline of results. In addition to the basic properties of

norms and traces in Chapter 1, we also establish a simple formula relating

H[1r]

and

H[7r],

whenever group

1r

is a quotient of group

7r,

say by relations

Wj(gf

1)

= 1,

where {gi} generate

7r

and {

Wj}

are words in the elements

gf1.

The formula is, as

might be guessed,

H[1r]

=

H[7r]/((wj- 1))

This formula makes it clear how a thorough understanding of

H[7r]

for free groups

7r

is important for understanding

H[1r]

for all finitely presented

1r.

We also prove in Chapter 1 that

H[1r]

and

TH[1r]

are finitely generated mod-

ules over a relatively small commutative ring, whenever

1r

is a finitely generated

group. Specifically, if

1r

is generated by

{gi},

1

~

i

~

n,

then

H[1r]

and

T

H[1r]

are

both finitely generated modules over the subring k[T(gi), T(gjgk)]

C

TH[1r],j k.

These traces are not algebraically independent if

n

~

4. The result proved in

Chapter 1 is improved substantially in later chapters, at least when

~ E

k, in the

sense that the number of module generators is greatly reduced and the relations

between generators are identified. Nonetheless, this early result shows that pass-

ing from the group rings

k[1r]

to the quotients

H[1r]

transports one from a type

of non-commutative algebra which is computationally intractable to a category of

much simpler non-commutative algebras, in which all computations reduce to linear

algebra over finitely generated commutative rings.

We insert a word here about the organization of the book. It seemed to the

authors that there was a certain central thread of development, which, although

technical, could still be followed without getting too bogged down with details. At

the same time, we wanted to include many other results and applications, which

were sometimes rather peripheral and often more technical. We have elected to

divide each of the chapters 1 through 9 in two parts. The first part of each chapter

is included in the main body of the text. The second parts are called Addenda

to the various chapters, and are relegated to the back of the book. Thus, at the

end of Chapter 1, the reader has a choice of going immediately to Chapter 2 or

looking for the Addendum to Chapter 1, to see what additional results are discussed.

Occasionally in the main text we refPr to a result from an addendum, but this

shouldn't cause any real problem continuing with the main text. Some of our

nicest results and applications are in the addenda, so we don't mean to downplay

their importance as part of the book. It is more a question of separating a first

level understanding from more specialized results.

In the Addendum to Chapter 1, we discuss some extra structure on the algebra

H[1r]

when

1r

is the fundamental group of the complement of a link in

S3

.

If the link

has d components then both the commutative ring T

H[1r]

and the algebra

H[1r]

are