Interactions complexe/symplectique : géométrie et dynamique. – SYMPLEXE

This project deals with the interaction between geometry and dynamics, which shows up as soon as there is some group action preserving some geometric structure. The groups that we have in mind are typically Lie groups, or lattices in Lie groups, or more generally, finitely generated groups. Several situations might occur: -- The group which is acting is not very interesting from the algebraic point of view, like for instance Z or R , but the action might be very rich. This is the realm of classical dynamical systems. This is notthe emphasis of our project. -- The geometric structure is rigid in the sense of Gromov, for instance a lorentzian metric, or a conformal structure, or a connection. This is also a rich and interesting research area, but again this is not the emphasis of our project. -- The group is highly non abelian, like a lattice in a simple Lie group, and the structure is flexible, like a symplectic structure, a complex structure, a contact structure etc. This is the main area that we want to investigate. As for the kind of results that we expect, we will focus on the interplay between dynamics and geometry, as explained below on some examples. These problems come from many different origins. Here is a sample: -- One of the first invariants of a geometry is its automorphism group (some people even don't distinguish between these two concepts). This is especially true when the group acts transitively. In some examples, it has been an extremely powerful tool in dynamics, like in the description of Anosov systems with smooth stable and unstable foliations by Ghys and Benoit-Foulon-Labourie. Here, the smoothness of the foliations generates a geometry which turns out to be homogeneous and leads to classification results. -- The trajectories of a holomorphic vector field, in C² for instance, are Riemann surfaces, typically with non abelian fundamental group. Their dynamics is directly described by the holonomy (pseudo) group which is holomorphic, and turns out to preserve geometric structures in good cases. --The rich and well established theory of group representations discusses homomorphisms from a given group to GL(n,C ). It is very tempting to analyze ``nonlinear representations as homomorphisms in a diffeomorphism group. Many examples are known, but almost everything remains to be done. Recently, important progresses have been realized in some very precise problems related to this (probably too vast) set of questions. These new results, together with our hopes to connect them, constitute the heart of our project. As a typical example of a central question, let us recall a (generalization of a) conjecture of Zimmer which attracted a lot of attention: Suppose that a lattice in a simple Lie group G acts faithfully by homeomorphisms on some compact manifold M. Does it follow that the dimension of M is larger than the real rank of G ? Some very special cases of this conjecture are known to be true: -- In dimension d=1 , the conjecture is settled for smooth actions of general lattices (Ghys1999, Burger-Monod 1999), and even for groups with Kazdhan's property T (Navas 2002). It is open for topological actions of general lattices. It has been proved for topological actions for some specific lattices (typically lattices commensurable to SL(n,Z) (n > 2) by Witte (1993)). -- In dimension 2, the conjecture is open in full generality. However, it has been proved by very different techniques for specific lattices (for instance lattices commensurable to SL(n,Z) with n > 2) under some additional assumptions: by Franks-Handel (2003) for smooth area preserving actions; by Polterovich (2002) for smooth area preserving actions on a closed oriented surface of genus at least 1; by Ghys (1993) for real analytic actions on closed surfaces different from the torus, by Deserti (2006) (graduate student in Rennes) for actions of SL(n,Z) (with n>3) for actions by birational transformations of the complex proje...