Abstract. We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (indgroup) if n ≥ 2. We describe the obstruction to this, which is of a topological nature.By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

Abstract. We study the normal subgroup f N generated by an element f = id in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On one hand if f has length at most 8 relatively to the classical amalgamated product structure of G, we prove that f N = G. On the other hand if f is a sufficiently generic element of even length at least 14, we prove that f N = G.

The automorphism group of the affine plane is a mysterious and challenging object. Although we know that it is an amalgamated product of two well known subgroups, many questions are still unsolved. Moreover, the group has the structure of an infinite-dimensional algebraic group. But the interactions between these two structures are not yet clear. In this paper, we study the length of an element (defined using the amalgamated structure) with respect to the algebraic structure. If the ground field is of characteristic zero, we prove that the length is a lower semicontinuous function on the group.

We study polynomial endomorphisms F of C N which are locally finite in the following sense: the vector space generated byWe show that such endomorphisms exhibit similar features to linear endomorphisms: they satisfy the Jacobian Conjecture, have vanishing polynomials, admit suitably defined minimal and characteristic polynomials, and the invertible ones admit a Dunford decomposition into "semisimple" and "unipotent" constituents. We also explain a relationship with linear recurrent sequences and derivations. Finally, we give particular attention to the special cases where F is nilpotent and where N = 2.

We give a family of polynomial automorphisms of the complex a ne plane whose generic length is 3 and degenerating in an automorphism of length 1 with surprisingly high degree.