On the graded quotients of the rings of Fricke characters of free groups

For a free group FnFn on n   generators, we consider the AutFn-invariant ideal J   generated by trx−2 for all x∈Fnx∈Fn in the ring XQ(Fn)XQ(Fn) of Fricke characters of FnFn. This ideal defines a descending filtration J⊃J2⊃J3⊃⋯J⊃J2⊃J3⊃⋯ of XQ(Fn)XQ(Fn). For each k≥1k≥1, we introduce the normal subgroup En(k)En(k) consisting of all automorphisms of FnFn which act on J/Jk+1J/Jk+1 trivially. These normal subgroups define a central filtration of AutFn. This is a Fricke character analogue of the Andreadakis–Johnson filtration An(k)An(k) of AutFn. The main purpose of the paper is to show that En(1)En(1) coincides with InnFn⋅An(2) where InnFn is the inner automorphism group of the free group FnFn, and that An(2k)⊂En(k)An(2k)⊂En(k) for any k≥1k≥1.