Amalgamations and automorphism groups

We review some facts about automorphism groups over polynomial rings. It is shown the the tame subgroup of the group of polynomials automorphisms of affine 3-space 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 in dimension 2. The result follows from the defining relations for the tame subgroup given by U. U. Umirbaev.