12
INTRODUCTION
set. For example, if 11"n is free on generators {gi}, 1 :::;
i :::;
n, then a representation
p: 11"n --+ SL(2, A) is just ann-tuple
hi
=
p(gi)} of determinant one matrices over
A, that is, a point of the variety SL(2,A)n
C
A4n. Now, it boggles the mind to
compare the simplicity of this observation with our view of a representation of 71"
n
as an involution preserving algebra homomorphism
I :
H(11"n] --+ M(2,
A),
where
the structure of H+[11"n] and H(11"n] has just taken about a hundred pages to work
out. Here are a few words in our defense.
First of all, the commutative ring
H+
[11" n] turns out to be what is known as
the affine coordinate ring of the character variety of the free group. The character
variety of a group 71" is the algebraic variety which is most closely related to the set of
conjugate classes of representations p: 71"--+ SL(2, K), where K is an algebraically
closed field, for example, K
=
C. The conjugate classes of representations are very
difficult to describe geometrically, even for the free group and K
=
C. Secondly, by
presenting an arbitrary group 71" as a quotient of a free group 11"n, one can describe
the set of representations
p:
71"--+ SL(2, A) as an algebraic subset of SL(2, A)n. One
just translates the group relations into polynomial relations between the 4n entries
of an n-tuple of matrices over A. Although this algebraic set of representations
of 71" is very easy to define in this manner, it is generally difficult to say much
about it, and even more difficult to say anything about the conjugate classes of
representations of 71". The gap between the H(11"] approach to representations and
the approach via algebraic sets narrows for non-free groups.
We continue this line of thought.
If
one writes down the relations between
the entries of an n-tuple of matrices implied by the factoring of a representation
p :
11"n --+ 71" --+ SL(2, A), one is led naturally to a finitely presented commutative
ring A(11"]
=
k[ai, bi,
ci, di]/
J,
where aidi- bici
=
1 and
J
is some ideal generated by
polynomials with integer coefficients. The commutative k-algebra A(11"] turns out
to be an invariant of 71", independent of presentation, which may contain nilpotent
elements. The points of the algebraic set of representations
p :
71" --+ SL(2, A), for
any k-algebra A, correspond bijectively to k-algebra homomorphisms ¢ : A(11"] --+
A.
The ring A(11"] is called the affine scheme of SL(2) representations of 71", and
is studied (in greater generality) in the book of Lubotzky and Magid (14], for
example. Because of the possibility of nilpotent elements in A(11"], it is a more
subtle invariant than, say, the algebraic set of SL(2, K) representations of 71", where
K is an algebraically closed field. The reduced ring A(1f]/
v'O
is the same thing as the
algebraic set. The A(11"] approach to representations resembles our H(11"] approach.
In fact, in Chapter 8 we study an action of G£(2, k) on the k-algebra A(11"],
related to conjugation of representations, and we study a related action of G£(2; k)
on the matrix ring M(2, A(11"]). Then in Chapter 9, we study the universal matrix
representation
P1r :
H(11"] --+ M(2, A(11"]), corresponding to the tautologous represen-
tation P1r : 71"--+ SL(2, A[11"]). The main result is the theorem of Procesi, mentioned
earlier, that if the ground ring
k
is a field of characteristic zero, then H+[11"] and
H[11"], respectively, map isomorphically to the rings of invariants A(1f]GL(
2,k)
C A(11"]
and M(2, A[11"])GL(2,k)
c
M(2, A(71"]). The same statement is true for arbitrary k if
71"
=
11"n is a free group, and this isomorphism H+[11"n]
=
A[11"n]GL(2,k)
C
A[11"n] is the
identification of H+[11"n] with the affine coordinate ring of the character variety of
the free group alluded to above.
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