Rings of Fricke characters and automorphism groups of free groups

Let F n be the free group on n free generators g~(v= 1, ..., n) and let ~ , be its group of automorphisms. Its quotient group (b* with respect to the inner automorphisms is the group of automorphism classes of F,. For n = 2, it is easy to derive a presentation for 45 since ~* is simply the linear group GL(2, 77) of 2 by 2 matrices with integers as entries and determinant __ 1. For n>3 , finding a presentation becomes a difficult problem which was first solved by Nielsen [16] and later by McCool [14] who also proved important theorems about the presentations of subgroups of ~,. However, not much is known about the structure of ~, and even less about that of ~b*. According to a general theorem of Baumslag [-1], ~b, is residually finite. It is already more difficult to prove the same result for ~* if n > 2 ; see Grossman [5]. Since there exists a natural mapping of ~, onto GL(n, ~) with a kernel which we shall denote by K,, it seems to be natural to concentrate ones attention on the structure of K, since GL(n, Z) is a well investigated group for all n. The quotient group of K, with respect to the inner automorphisms shall be called K*. We know [11] that K, (and, therefore, K*) are finitely generated with explicitly known generators. We also know that K, is residually torsion free nilpotent. This follows immediately from the action of K n on the group ring of F, which is a graded ring [-10] in which the powers of the augmentation ideal provide the grading. The action of K, on this ring then provides a faithful representation of K, in terms of upper triangular (infinite) matrices with integers as entries and with terms + 1 in the main diagonal. But for K* it is not even known whether it is torsion free or not. Nor is it known whether K, (and, therefore, K*) has a finite presentation. Also, no finite dimensional matrix representation for ~b,, n > 2 or ~*, n > 3, are known and there is at least some support for the conjecture that none exist. (It would be sufficient to show this for ~2; see [13].) However, it has been known for a long time ~4] that ~* acts as a group of automorphisms of a quotient ring Rn of a finitely generated (commutative) ring and it has been shown later [7] that this