Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)

We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.Comment: The first version of this paper and "A note on integrability and finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate version re-submitted to the journal on March 2015 is available at http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a comparison between these 3 version

Finite Groups of Essential Dimension 2

We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0.Comment: 30 pages (To appear in Commentarii Mathematici Helvetici

The Galois group of random elements of linear groups

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of \Gamma (but not on \Gamma)