Co-tame polynomial automorphisms

A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms, including $3$-triangular automorphisms, are co-tame. Of particular interest, if $n=3$, we show that the statement "Every $m$-triangular automorphism is either affine or co-tame" is true if and only if $m \leq 3$; this improves upon positive results of Bodnarchuk (for $m \leq 2$, in any dimension $n$) and negative results of the authors (for $m \geq 6$, $n=3$). The main technical tool we introduce is a class of maps we term 'translation degenerate automorphisms'; we show that all of these are co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.