On the closure of the tame automorphism group of affine three-space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of \C^3 is not closed, when the latter is seen as an infinite dimensional algebraic group.tomorphism group of affine three-spac

Remarks on a normal subgroup of GA_n

We show that the subgroup generated by locally finite polynomial automorphisms of k^n is normal in GA_n. Also, some properties of normal subgroups of GA_n containing all diagonal automorphisms are given.Comment: 5 page

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates

The tame and the wild automorphisms of an affine quadric threefold

We prove the existence of wild automorphisms on an affine quadric threefold. The method we use is an adaptation of the one used by Shestakov and Umirbaev to prove the existence of wild automorphisms on the affine three dimensional space.Comment: Minor corrections. To appear in Journal of the Mathematical Society of Japa

Automorphism Groups of Configuration Spaces and Discriminant Varieties

The configuration space $\mathcal{C}^n(X)$ of an algebraic curve $X$ is the algebraic variety consisting of all $n$-point subsets $Q\subset X$. We describe the automorphisms of $\mathcal{C}^n(\mathbb{C})$, deduce that the (infinite dimensional) group Aut$\,\mathcal{C}^n(\mathbb{C})$ is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of $\mathcal{C}^n(\mathbb{C})$ are also computed. We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level. This is an extended version of our paper \cite{Lin-Zaidenberg14}. We strengthened the results concerning the automorphism groups of cylinders over rigid bases, replacing the rigidity assumption by the weaker assumption of tightness. We also added alternative proofs of two auxiliary results cited in \cite{Lin-Zaidenberg14} and due to Zinde and to the first author. This allowed us to provide the optimal dimension bounds in our theorems.Comment: 61p.; an acknowledgment added; see also : V. Lin and M. Zaidenberg, Configuration spaces of the affine line and their automorphism groups In: Automorphisms in Birational and Complex Geometry. Ivan Cheltsov et al. (eds.), 431-468. Springer Proceedings in Mathematics and Statistics, vol. 79, 201