On the Graded Quotients of the Algebra of SL(2) Characters of Free Groups

Let F n be a free group of rank n. An SL ( 2 ) character of F n means the trace of an SL ( 2 ) representation of F n. Let K be a field of characteristic 0. The automorphism group Aut ( F n ) naturally acts on the commutative K-algebra of the SL ( 2 ) characters. Then the augmentation ideal J n + of the commutative K-algebra generated by the SL ( 2 ) characters is an Aut ( F n )-invariant. The main purpose of this paper is to study the structure of the graded quotient ( J n + ) k / ( J n + ) k + 1 as an Aut ( F n )-module.