The tame and the wild automorphisms of an affine quadric threefold (Q1946858)

In the landmark paper [J. Am. Math. Soc. 17, No. 1, 197--227 (2004; Zbl 1056.14085)], \textit{I. P. Shestakov} and \textit{U. U. Umirbaev} proved that there exist some wild automorphisms in \(\mathrm{Aut}(\mathbb{A}^3)\), which are defined as automorphisms that cannot be written as a composition of finitely many triangular and affine automorphisms. In this paper under review, the authors show an analogous result about the automorphism \(\mathrm{Aut}(\mathrm{SL}_2(\mathbb{C}))\). The tame subgroups of \(\mathrm{Aut}(\mathrm{SL}_2(\mathbb{C}))\) is defined to be the group generated by elementary automorphisms and \(\mathrm{O}_4(\mathbb{C})\). An automorphism that is not in the tame subgroup is called wild. The authors also define a technical notion of an elementary reduction. Then the main result of this paper is that every tame automorphism of \(\mathrm{SL}_2(\mathbb{C})\) admits a sequence of elementary reductions to an element of \(\mathrm{O}_4(\mathbb{C})\). Using the main result, the authors are able to produce some wild automorphisms on \(\mathrm{SL}_2(\mathbb{C})\). At the end of the paper, the authors show how to adapt this construction to the case of \(\mathrm{PSL}_2(\mathbb{C})\).