Preprint C01/2001
Groups of germs of analytic diffeomorphisms

Fabio Enrique Brochero Martinez

**Keywords: **
analytic germ | convergent orbits | formal and analytic conjugation

The study of germs of holomorphic diffeomorphisms and finitely
generated groups of analytic germs of diffeomorphisms fixing the
origin in one complex variable was started in the XIX century, and
has been intensively studied by mathematicians
in the past century.
We deal with germs of
diffeomorphisms in $(\C^2,0)$ and the finitely generated group of
germs
First, we study the finite groups of diffeomorphisms. We prove a
generalization of Mattei-Moussu topological criteria about finiteness of a
group.we compare a topological conjugacy
class of some finite diffeomorphisms with their analytical conjugacy class,
i.e. we construct the moduli space (topological vs analytical) of the diffeomorphisms that are conjugate with some
finite order diffeomorphism.
Next, we study the groups of diffeomorphisms supposing
that they have some algebraic structure. We prove that if $\Cal
G\subset \dihdos{}$ is a solvable group then its $7^{th}$
commutator subgroup is trivial. Furthermore, we characterize the
abelian subgroup of diffeomorphisms tangent to the identity, and
in the case when the group contains a dicritic diffeomorphism,
i.e. the group contains a diffeomorphism
$F(X)=X+F_{k+1}(X)+\cdots$ where $F_{k+1}(X)=f(X)X$ and $f$ is a
homogeneus polynomial of degree $k$, we prove that the group is a
subgroup of a one parameter group. We write $\dihdos{_1}$ to
denote the group of diffeomorphisms tangent to the identity at
$0\in \C^2$.
We analyze the behavior of the orbits of a
diffeomorphism tangent to the identity. We prove a
generalization of the one dimensional flower theorem to two
dimensional dicritic diffeomorphisms.
Finally, we show the formal classification of diffeomorphisms
tangent to the identity using the notion of the semiformal conjugacy.
We show that a representative diffeomorphism found using
semiformal conjugacy and a cocycle determine its formal conjugacy class.