Normal subgroup generated by a plane polynomial automorphism

We study the normal subgroup 〈f〉 _N generated by an element f ≠ id in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On the one hand, if f has length at most 8 relative to the classical amalgamated product structure of G, we prove that 〈f〉 _N = G. On the other hand, if f is a sufficiently generic element of even length at least 14, we prove that 〈f〉 _N ≠ G.