Approximate tameness in free associative and free Poisson algebras (n ≥ 3) and status of Anick-type automorphisms

Determine whether every automorphism of the free associative algebra K⟨x1,…,xn⟩ and the free Poisson algebra in n ≥ 3 variables over a field of characteristic zero is approximately tame with respect to the formal power series topology; equivalently, establish an analogue of the Shafarevich–Anick approximate tameness theorem for these algebras, and in particular ascertain whether the Anick automorphism of K⟨x,y,z⟩ and the Shestakov–Zhang Anick-type automorphism of free Poisson algebras are approximately tame.

Background

Shafarevich and Anick proved that every automorphism of the polynomial algebra K[x1,…,xn] over a field of characteristic zero is approximately tame, i.e., can be approximated by tame automorphisms in the formal power series topology. For free associative and free Poisson algebras, the corresponding analogue is not established, and the paper introduces tangent Lie algebras and the formal power series topology to paper tameness and wildness in these settings.

The authors highlight specific wild automorphisms—the Anick automorphism in K⟨x,y,z⟩ and its Poisson analogue constructed by Shestakov and Zhang—and explicitly note that it is unknown whether these are approximately tame, motivating the broader question for all automorphisms in n ≥ 3 variables.