The Strong Anick Conjecture is true

Abstract

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra $K⟨x,y,z⟩$ over a field $K$ of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of $K⟨x,y,z⟩$. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of $K⟨x,y,z⟩$. We also find a large new class of wild automorphisms of $K⟨x,y,z⟩$ which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results.