Erroneous proofs of the wildness of some automorphisms of free metabelian Lie algebras

Abstract

The well-known Bachmuth-Mochizuki-Roman'kov Theorem \cite{BM,Romankov85} states that every automorphism of the free metabelian group of rank $\geq 4$ is tame. In 1992 Yu. Bahturin and S. Nabiyev \cite{BN} claimed that every nontrivial inner automorphism of the free metabelian Lie algebra $M_n$ of any rank $n\geq 2$ over a field of characteristic zero is wild. More examples of wild automorphisms of $M_n$ of rank $n\geq 4$ were given in 2008 by Z. \"Ozcurt and N. Ekici \cite{OE}. The main goal of this note is to show that both articles contain uncorrectable errors and to draw the attention of specialists to the fact that the question of tame and wild automorphisms for free metabelian Lie algebras $M_n$ of rank $n\geq 4$ is still widely open.