Some Properties Of The Cremona Group Julie Dserti by Julie Déserti instant download after payment.

Abstract. We recall some properties, unfortunately not all, of the Cremona
group.
We first begin by presenting a nice proof of the amalgamated product
structure of the well-known subgroup of the Cremona group made up of the
polynomial automorphisms of C2. Then we deal with the classification of
birational maps and some applications (Tits alternative, non-simplicity...)
Since any birational map can be written as a composition of quadratic
birational maps up to an automorphism of the complex projective plane,
we spend time on these special maps. Some questions of group theory are
evoked: the classification of the finite subgroups of the Cremona group and
related problems, the description of the automorphisms of the Cremona
group and the representations of some lattices in the Cremona group.
The description of the centralizers of discrete dynamical systems is an
important problem in real and complex dynamic, we describe the state of
the art for this problem in the Cremona group.
Let S be a compact complex surface which carries an automorphism f
of positive topological entropy. Either the Kodaira dimension of S is zero
and f is conjugate to an automorphism on the unique minimal model of
S which is either a torus, or a K3 surface, or an Enriques surface, or S is
a non-minimal rational surface and f is conjugate to a birational map of
the complex projective plane. We deal with results obtained in this last
case: construction of such automorphisms, dynamical properties (rotation
domains...).