This research proposal is a continuation of the previous project with the title "Automorphism Groups of Varieties: Geometry, Combinatorics, and Representations". The main object is the automorphism group Aut(X) of an affine algebraic variety X, i.e. the group of regular automorphisms of X. A lot is known for curves X, but almost nothing in higher dimension. The only case studied more carefully is the affine Cremona group Aut(An), the automorphism group of affine n-space An = Cn. In particular, the automorphism group of the plane A2 got a lot of attention in recent years; it also appears in some of our research projects.

One of the fundamental questions can be expressed as follows.

Basic Problem. How much information about the structure of the affine variety X can be retrieved from the automorphism group Aut(X)?

The group Aut(An) has the structure of an ind-variety, i.e. an infinite dimensional variety in the sense of Shafarevich. This group will serve as a model and “test case” for our general studies. We have recently shown that for every affine variety X the automorphism group Aut(X) admits a canonical structure of an ind-group.

One of the main tools is to study the Lie algebra Lie Aut(X) which is sitting in the Lie algebra Vec(X) of vector fields on X. A basic question is the relation between closed connected subgroups G ⊂ Aut(X) and their Lie algebra Lie G ⊂ Lie Aut(X). For example, the Lie algebra Vec0(An) of vector fields of divergence 0 is simple, but we do not know if the corresponding group SAut(An) of automorphisms of Jacobian determinant equal to 1 contains closed normal subgroups or not.

Already Aut(A2) is a very challenging object. Our knowledge of this group is mainly based on the fact that Aut(A2) has the structure of an amalgamated product. However, this structure does not generalize to higher dimension n > 2. This “explains” the difference between dimension 2 and dimension > 2.