Bodnarchuk Y. Infinite-dimensional algebraic polynomial transformation groups of affine spaces.

The structure of closed subgroups of the affine Cremona group over an algebraically closed field of zero characteristic, which contain a special linear subgroup is ascertained. It is proved that an ordinary affine group is a maximal closed (in Ind- Zariski's topology) subgroup of the affine Cremona group. Similar results are obtained for the polynomial transformations group of a symplectic space. Over an arbitrary field of zero characteristic it is shown that a group which contains an affine group and at least one a nonlinear polynomial map acts k-ply transitively on the affine space for an arbitrary k chosen in advance. These results can be considered as algebraic analogies of B.Mortimer's theorem which asserts that "nearly always" a finite affine group is a maximal subgroup in a correspondent symmetrical group. As regards to the maximality of the affine group as an abstract subgroup, it is shown that in the dimension n>2, the affine group together with an arbitrary nonlinear triangular map generatethe group of tame invertible polynomial maps. For n=2 this assertion is wrong. Some classes of a small compose-triangular length maps, each of them together with an the affine group generate are indicated. The well known Peter Neumann theorem about isomorphisms of standard wreath products of groups is generalized on wreath products of arbitrary transitive transformation groups with abstract groups. Regular automorphisms of block-unitraangular and block-triangular polynomial translation groups are described by the wreath products calculation technique. It is proved that over a field of zero characteristic all regular automorphisms of block-triangular polynomial translation groups (in particular Jonquier's group) are inner ones. It is ascertained that such the groups over finite fields have outer automorphisms, moreover all automorphism group is a semidirect product of elementary abelian p-group and the subgroup of interior automorphisms. It is proved that all regular automorphisms of the affine Cremona group over an algebraic closed field of zero characteristic is inner.