The strong Anick conjecture

Abstract

Recently, Umirbaev 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 work, 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 discover a large, previously undescribed class of wild automorphisms of K(x, y, z) that is not covered by the results of Umirbaev. © 2006 by The National Academy of Sciences of the USA.