Семинар НУЛ АГП «On the topological simplicity of the automorphism group of affine space»

In affine geometry, one of the central objects of study is the group Aut(n) — the automorphism group of n-dimensional affine space. Despite extensive research, many natural questions about this group remain open for n ≥ 3. For instance, in dimensions 3 and higher, no explicit generating set is known. In contrast, the automorphism group of the affine plane is much better understood. As early as 1942, Jung proved that the automorphism group of the complex affine plane is generated by tame automorphisms, and in 1953, van der Kulk extended this result to fields of positive characteristic.  

Alongside the entire automorphism group Aut(n), its normal subgroup SAut(n) — consisting of elements with Jacobian determinant equal to 1 — has been actively studied. Following the same trend, while the question of whether SAut(n) is simple as an abstract group for n ≥ 3 remains open to this day, Danilov already obtained a nontrivial normal subgroup in SAut(2) back in 1973. Given that Aut(n) and SAut(n) carry a natural structure of an ind-group, one can similarly ask whether they are topologically simple, i.e., whether they contain any nontrivial closed normal subgroups.  

In his 1981 paper describing the ind-group structure, Shafarevich claimed a proof of the topological simplicity of SAut(n) for any n over fields of characteristic zero. However, a mistake was later discovered in the proof, leaving the question open once again — even in dimension 2.

In this talk, following Blanc’s paper (2024), we will prove the topological simplicity of SAut(2) over infinite fields and take the first step toward proving a similar result in higher dimensions. We will also discuss the question about closed normal subgroups of Aut(n) over infinite fields and, if time permits, touch on the case of finite fields.