Groups of automorphisms of some affine varieties

In 1966 Shafarevich introduced the notion of an ind-variety. It turns out that Aut(X) has a natural structure of an ind-variety for any affine algebraic variety X. In this thesis we study the structure of Aut(X) viewed as an ind-group
and a structure of a Lie algebra Lie Aut(X). We compute the automorphism group of the Lie algebra of the group of automorphisms of an affine n-space (jointly with Hanspeter Kraft). We also prove that Lie subalgebras of Lie Aut(A^2) isomorphic to
the Lie algebra of the group of affine transformations of an affine plane A^2 are isomorphic if and only if Jacobian Conjecture holds in dimension 2. In the second part of the thesis we consider an n-dimensional affine variety X endowed with a non-trivial regular SL(n,C)-action. We prove that if Aut(X) is isomorphic to Aut(Y) as an ind-group for some irreducible affine normal variety Y, then Y is isomorphic to X as a variety. At the end of the thesis we present an example found with Matthias Leuenberger of two affine surfaces such that their so-called special automorphism groups are isomorphic as abstract groups, but not isomorphic as ind-groups.