INTRODUCTION 5
H[1r] is justified. In a way, what we are rediscovering is the fact that the Cayley-
Hamilton Identity for 2 x 2 matrices implies all other matrix and trace identities.
This brings up another point. If one wants to verify a matrix identity, one can
plug in matrices with indeterminate coefficients and check the identity by carrying
out all the multiplications and additions which occur in the identity. This is often
very strenuous. All that arithmetic with the matrix entries fills pages, and seems
rather disjointed. It is better if the matrices can be manipulated as single variables,
obeying certain basic laws which then are seen to imply more subtle laws. This is
similar to what we are trying to do in H[1r]. Instead of studying representations
p :
1r ----. SL(2, A) by choosing a bunch of matrices
p(gi)
and doing a lot of arithmetic
with the matrix entries, corresponding to relations between the 9i E 1r, we do as
much arithmetic as possible in advance in H[1r], where
T(g)
=
g
+
g-
1
is central,
all
g E
1r. We are then in a better position to see more coherence and structure
in the representations
p:
H[1r] ----.
M(2,
A). Somehow, standard techniques such as
choosing bases of
A
2
and putting certain matrices in normal form, are pushed to
the background. In fact, these techniques don't even work if
A
is not a field. In
our approach,
A
is irrelevant until the very end, and certainly does not need to be
a field.
But we digress. Let's return to the outline of Chapter 2. We introduce the
notion of a trace ideal I
c
H[1r], which just means that I is invariant under the
trace
T(x)
=
x
+
L(x),
or equivalently under the involution
i.
If
!
E k then
I= I+
EB
I-
c
H+[1r] EB H-[1r]. If
I= ((x1))
is a two-sided trace ideal and if 1r is
generated by
{gi}
then we show
I+=
(xj,
I;~
I,
ix1gkgti)
c
H+[1r].
The most important example is
I= ((w1 -
1)), where
w1
E
71".
Then
Ht]
=
H[7r] and
H;11r]
=
H+[7r]
where 7T is the quotient group of 1r obtained by dividing by relations w1 = 1. Once
we have finite presentations of H[7T] and H+[n] for free groups
7T,
then we get from
these observations finite presentations of H[1r] and H+[1r] for any finitely presented
group 1r. We also get by this method presentations of H-[1r], as H+[1r]-module.
It is curious that when
!
~
k,
we do not know how to write down a presentation
of TH[1r]. This is even the case for most two-generator groups 1r, where we know
everything about H[1r2] and T H[1r2] for the free group 1r2 on two generators.
We now come to chapters 3 and 4, in which we study groups 1r with two gen-
erators {a, b }. In Chapter 3, we study the structure of H[1r], and in Chapter 4,
we study absolutely irreducible group representations
p :
1r----. S£(2, R) via algebra
representations
p :
H[1r] ----.
M(2,
R). A representation
p :
1r ----. SL(2, R) is abso-
lutely irreducible if the image
p(1r)
C
SL(2, R)
C M(2,
R) spans
M(2,
R) over R.
The ring
R
need not be a field. We are able to carry out much of the two-generator
group study over arbitrary ground rings, that is, we allow
!
~
k. On the other
hand, if
!
E
k then we get additional results. We introduce quaternion algebras
into the discussion, and prove that certain localizations of H[1r] are quaternion al-
gebras over localizations of H+[1r]. For certain groups, for example two-bridge knot
groups, H[1r] itself is a quaternion algebra over H+ [1r], without any localization.
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