The Amalgamated Product Structure of the Tame Automorphism Group in Dimension Three

It is shown the the tame subgroup $\text{TA}_3(\mathbb C)$ of the group $\text{GA}_3(\mathbb C)$ of polynomials automorphisms of ${\mathbb C}^3$ can be realized as the product of three subgroups, amalgamated along pairwise intersections, in a manner that generalizes the well-known amalgamated free product structure of $\text{TA}_2(\mathbb C)$ (which coincides with $\text{GA}_2(\mathbb C)$ by Jung's Theorem). The result follows from defining relations for $\text{TA}_3(\mathbb C)$ given by U. U. Umirbaev.