Abstract:
In affine geometry, one of the central objects of study is the group $\mathrm{Aut}(n)$ —
the automorphism group of $n$-dimensional affine space. Despite extensive
research, many natural questions about this group remain open for $n \ge 3$. 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 $\mathrm{Aut}(n)$, its normal subgroup $\mathrm{SAut}(n)$ —
consisting of elements with Jacobian determinant equal to $1$ — has been
actively studied. Following the same trend, while the question of whether
$\mathrm{SAut}(n)$ is simple as an abstract group for $n \ge 3$ remains open to this day,
Danilov already obtained a nontrivial normal subgroup in $\mathrm{SAut}(2)$ back in 1973.
Given that $\mathrm{Aut}(n)$ and $\mathrm{SAut}(n)$ 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 $\mathrm{SAut}(n)$ 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 $\mathrm{SAut}(2)$ over infinite fields and take the first step toward
proving a similar result in higher dimensions. We will also discuss the
question of topological simplicity for $\mathrm{Aut}(n)$ over infinite fields and, if
time permits, touch on the case of finite fields.
