Final Report Summary - FLAT SURFACES (SL(2,R)-action on flat surfaces and geometry of extremal subvarieties of moduli spaces)
Flat surfaces are complex surfaces with a flat metric. The interest in flat surfaces stems from the dynamics of billiards. Indeed, gluing two copies of usual rectangular billiard tables back to back one obtains the simplest example of a flat surface.
The dynamics on flat surfaces is measured by two important quantities, Lyapunov exponents and Siegel-Veech constants. The Lyapunov exponents measure roughly the topology of a very long billiard trajectory. The Siegel-Veech constants measure the asymptotic number of closed trajectories up to a given length bound. Individual Lyapunov exponents are very hard to calculated and next to nothing is known about their number-theoretic properties.
Major progress towards understanding these quantities has been made in the project. The sums of Lyapunov exponents are more tractable invariants and M.Kontsevich and A.Zorich had observed a non-varying phenomenon a decade ago. In joint work with D.Chen we explained this non-varying phenomenon through methods of algebraic geometry. While an effective computation of Lyapunov exponents is still out of reach, we obtained with C.Matheus and J.C.Yoccoz an algorithmic criterion to prove the simplicity of the Lyapunov spectrum.
The method of Lyapunov exponents has also been successfully applied to a classification problem in a different context. Non-arithmetic ball quotients are highly interesting symmetric manifolds. All known constructions are equivalent to the Deligne-Mostow examples. But it was an open problem to detect how many essentially different examples they have constructed. With A.Kappes we have shown that Lyapunov exponents can be used to tell apart the different cases.
A long term goal is to pass from finite type flat surfaces to infinite type. As a stepping stone towards this, A.Eskin and A.Zorich had made experiments for the large genus asymptotic behavior of Siegel-Veech constants. In joint work with D.Zagier we have investigated the arithmetic nature of Siegel-Veech constants for special types of flat surfaces. The generating series of these Siegel-Veech constants turned out to be very nice, they are quasimodular forms. This is one of the new observations that we used to prove the large genus conjecture.
Among all flat surfaces, those with optimal dynamics are of particular interest. They give rise to so-called Teichmüller curves. There are very few series of Teichmüller curves known presently and, consequently, the classification of Teichmüller curves is a major problem in this field. We contributed to this problem by an effective finiteness result (joint with M.Bainbridge and P.Habegger). Improving the a priori bounds and turning this criterion into a feasible algorithm is one of the challenges for the future. Several Ph.D. projects in the research group have contributed to determining the basic invariants of Teichmüller curves and towards the extension of the finiteness results to more cases.
Modular curves curves have been investigated for decades because of their interesting interaction of arithmetic and geometric properties. Teichmüller curves are not modular curves, but they possess a special map, called modular embedding, that makes them clear candidates for the next most interesting class of curves with arithmetic flavor. In joint work with D.Zagier we make this precise, by actually computing these modular embeddings. This result should be thought of as the starting point to construct and study curves with modular embeddings independently of the connection to flat surfaces.
The dynamics on flat surfaces is measured by two important quantities, Lyapunov exponents and Siegel-Veech constants. The Lyapunov exponents measure roughly the topology of a very long billiard trajectory. The Siegel-Veech constants measure the asymptotic number of closed trajectories up to a given length bound. Individual Lyapunov exponents are very hard to calculated and next to nothing is known about their number-theoretic properties.
Major progress towards understanding these quantities has been made in the project. The sums of Lyapunov exponents are more tractable invariants and M.Kontsevich and A.Zorich had observed a non-varying phenomenon a decade ago. In joint work with D.Chen we explained this non-varying phenomenon through methods of algebraic geometry. While an effective computation of Lyapunov exponents is still out of reach, we obtained with C.Matheus and J.C.Yoccoz an algorithmic criterion to prove the simplicity of the Lyapunov spectrum.
The method of Lyapunov exponents has also been successfully applied to a classification problem in a different context. Non-arithmetic ball quotients are highly interesting symmetric manifolds. All known constructions are equivalent to the Deligne-Mostow examples. But it was an open problem to detect how many essentially different examples they have constructed. With A.Kappes we have shown that Lyapunov exponents can be used to tell apart the different cases.
A long term goal is to pass from finite type flat surfaces to infinite type. As a stepping stone towards this, A.Eskin and A.Zorich had made experiments for the large genus asymptotic behavior of Siegel-Veech constants. In joint work with D.Zagier we have investigated the arithmetic nature of Siegel-Veech constants for special types of flat surfaces. The generating series of these Siegel-Veech constants turned out to be very nice, they are quasimodular forms. This is one of the new observations that we used to prove the large genus conjecture.
Among all flat surfaces, those with optimal dynamics are of particular interest. They give rise to so-called Teichmüller curves. There are very few series of Teichmüller curves known presently and, consequently, the classification of Teichmüller curves is a major problem in this field. We contributed to this problem by an effective finiteness result (joint with M.Bainbridge and P.Habegger). Improving the a priori bounds and turning this criterion into a feasible algorithm is one of the challenges for the future. Several Ph.D. projects in the research group have contributed to determining the basic invariants of Teichmüller curves and towards the extension of the finiteness results to more cases.
Modular curves curves have been investigated for decades because of their interesting interaction of arithmetic and geometric properties. Teichmüller curves are not modular curves, but they possess a special map, called modular embedding, that makes them clear candidates for the next most interesting class of curves with arithmetic flavor. In joint work with D.Zagier we make this precise, by actually computing these modular embeddings. This result should be thought of as the starting point to construct and study curves with modular embeddings independently of the connection to flat surfaces.