On the augmentation topology of automorphism groups of affine spaces and algebras

We study topological properties of Ind-groups Aut(K[x₁,⋯,x_n]) and Aut(K(x₁,⋯,x_n)) of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of Aut(Aut(A)), where A is the polynomial or free associative algebra over the base field K. We prove that all Ind-scheme automorphisms of Aut(K[x₁,⋯,x_n]) are inner for n ≥ 3, and all Ind-scheme automorphisms of Aut(K(x₁,⋯,x_n)) are semi-inner. As an application, we prove that Aut(K[x₁,⋯,x_n]) cannot be embedded into Aut(K(x₁,⋯,x_n)) by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov-Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.