Geometry and Dynamics of the Moduli Space – GeoDyM

The project “Geometry and Dynamics of the Moduli space” is a project in Mathematics. Having a core in dynamics, this project is located at the frontier between dynamics, geometry, algebraic geometry, topology, combinatorics, and representation theory. Certain classical problems of one-dimensional dynamics, of dynamics on surfaces, of billiards in polygons, of surface foliations, can be solved using the recent knowledge about the moduli spaces. Reciprocally, dynamical and geometric methods produce results about the moduli spaces which were are difficult to obtain through pure algebro-geometric or complex-analytic approaches.

We list the concrete problems (organized as tasks and subtasks) that form the core of the project:
1. Square-tiled surfaces and Veech surfaces including continued fraction algorithms, Rauzy-Veech induction and Rauzy diagrams, applications of representations of finite groups, arithmetic Teichmuller discs, expanders.
2. Lyapunov exponents, including Lyapunov exponents for cyclic covers and arithmetic Teichmuller discs, geometric compactification of strata of the moduli space of Abelian differentials, asymptotic behaviour of the determinant of flat Laplacian, and Siegel-Veech constants.
3. Spectrum of the leafwise hyperbolic Laplacian, including study of analytic and asymptotic properties the Ruelle zeta-function.
4. Asymptotics when genus tends to infinity and infinite translation surfaces, including ergodic properties of infinite translation surfaces, diffusion and “Wind Tree” model, asymptotics of geometric and dynamical characteristics of strata when genus tends to infinity.
5. Small dilatation problem and geometry and topology of the strata and moduli, including study of systols of the strata, applications of technique in 3-dimensional manifolds, study of the Lipshitz metric, of train-track automata and of the fundamental group of the strata.

The researchers are grouped into 3 partners: Rennes, Marseille, Paris.

The partner “PARIS” includes world experts in dynamics as A.Avila, R. Krikorian, J.C.Yoccoz and the group around them on one hand, and the world expert in geometry as M.Kontsevich on the other hand.

The partner “MARSEILLE” involves a group of leading experts in combinatorial aspects of dynamics as P.Arnoux and S.Ferenczi; this group complement the Paris group in dynamics. Marseille partner also has an excellent group in geometric dynamics, represented by C.Boissy, P. Hubert, E. Lanneau, J. Los; this group complements and relates the first two. Finally, this partner has a fantastic experimental team around V. Delecroix, T. Monteil (with agents in Paris like S. Lelievre and in Rennes like M.Bauer).

The partner “RENNES” has an excellent team specialising in geometry represented by A.Lenzhen and by B.Wiest (this team complements algebraic geometry from Paris and geometric dynamics from Marseille). This partner also has a strong dynamical group around S.Gouezel and A.Zorich, which will take care of spectral aspects (Laplacian, Lyapunov exponents) to complement Paris and Marseille dynamical groups.

The problems which we plan work on are very ambitious and require consolidation of different approaches. We hope that the well-coordinated skills dispatched between different groups and partners will allow us to attack these problems on a large front.

The necessary condition of the success of the project is a constant contact with world experts in the area. We plan to invite the specialists of our research area and visit them as often as possible. Several projects of the members of the team involving leading international experts in our area from USA, Germany, Italy are already in progress.

We plan to organize two extended ANR team meetings and a broad international main conference preceded by a summer school.