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
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