6
INTRODUCTION
We study two-bridge knot groups extensively in the Addendum to Chapter 4, re-
formulating some of the results of Riley on non-abelian
SL(2)
representations of
these groups, [21].
First in Chapter 3, for the free group
1r2
on
{a,
b} we show that
T
H[1r2]
is a
polynomial ring on three generators,
TH[1r2]
=
k[T(a), T(b), T(ab)]
and we show
that
H[1r2]
is a free module of rank 4 over
TH[1r2],
with basis
{1,
a, b, ab}.
We also
write down the (non-commutative) multiplication table for these basis elements.
The fact that the elements listed generate
T
H[1r2]
and
H[1r2]
is an easy consequence
of formulas from Chapter 1. We prove the polynomial ring statement about
T
H[1r2]
by exploiting a certain representation
p :
1r2
--+
SL(2, k[x,
y,
z]),
where
x,
y,
z
are
indeterminates over k. It is only necessary to arrange that the traces tr(p(a)),
tr(p(b)),
tr(p(ab))
E
k[x,
y, z] are algebraically independent. We prove the free
module statement about
H[1r2]
by exploiting the trace form
T( uv) :
H[1r2]x H[1r2]
--+
TH[1r2].
A module with a form is necessarily a free module if an appropriate
determinant associated to the form is not a zero-divisor. We use this fact many
times throughout the paper.
For any group
1r
with two generators, we give a sufficient condition for
H[1r]
to
be free ofrank 4 over
TH[1r].
Let
A= T(a), B
=
T(b), C
=
T(ab)
E
TH[1r].
Then
the condition is that
A
2
+
B
2
+
C
2
-
ABC-
4
E
T
H[1r]
is not a zero-divisor. More
generally, if¢ :
TH[1r]
--+
R
is a ring homomorphism with
R
commutative, then
R
®
H[1r]
is free of rank 4 over
R
if ¢(A2
+
B
2 +
C
2
-ABC-
4)
E
R
is not a
TH[1r]
zero-divisor. The element A2
+
B
2
+
C
2
-
ABC-
4 will be familiar to anyone who
has computed the trace of a commutator in
SL(2).
In fact,
If¢:
TH[1r]
--+
R
is a homomorphism such that
¢(T(aba- 1b-
1
)-
2) E
R
is
not a zero-divisor then we prove in Chapter 4 that ¢ extends to a representation
iP:
H[1r]
--+
M(2, R'),
where
R'
is either
R
or a quadratic extension of
R.
Moreover,
the group representation p:
1r--+ SL(2, R')
associated to
iP
is absolutely irreducible
if and only if
¢(T(aba- 1b-
1
)-
2)
=
tr(p(aba- 1b-
1
))-
2
E
R'
is a unit. If
R'
is a
field then the conjugacy class of p, or of
iP
= p,
is uniquely determined by the trace
homomorphism¢ :
TH[1r]
--+
R'
(assuming still
¢(T(aba- 1b-
1
))
=/:-
2).
Results
relating absolutely irreducible representations p:
1r--+ SL(2, K)
with the condition
tr(p(
aba -
1
b-
1
))
=/:-
2 are well-known when
K
is a field. One can find a discussion
in the paper of Culler and Shalen [5], for example.
Our philosophy is that
H[1r]
is an interesting invariant of groups
1r,
which en-
codes all information about
SL(2)
representations of
1r,
and which has a beautiful
internal structure of its own. Many of our results, especially those concerning repre-
sentations p:
1r--+ SL(2, K)
where
K
is a field, are reformulations of known results.
This becomes even more true after chapters 8 and 9, where we identify H+
[1r]
and
H[1r]
with rings defined in terms of
GL(2,
k)-invariant subrings of polynomial rings
and matrix rings. But we believe some of our results on
H+[1r]
and
H[1r]
in chapters
5 and 6 give new information about these classical rings of invariants for groups
with four or more generators. The results in chapters 3 and 4 on two generator
groups and their absolutely irreducible representations are meant partly to be a
warm-up for harder results in later chapters. Also, in the end, the most delicate
results about
H[1r]
reflect properties of representations p:
1r--+ SL(2, A)
which are
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