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Flat surfaces. (English) Zbl 1129.32012
Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-23189-9/hbk). 437-583 (2006).
This is a nice survey paper on an interesting subject, namely, flat surfaces with cone singularities (called flat surfaces for short ). The author presents several ideas, results (old and recent) and open questions in the field. Particular attention is paid to the most important notions, such as the behaviour of geodesics on flat surfaces, the study of the dynamics and the ergodic properties of the geodesic flow associated to such a surface, problems having to do with counting closed geodesics, the structure and the stratification of moduli spaces of flat surfaces, the geodesic flow of moduli spaces, the volumes of these moduli spaces, the relation of flat structures with billiards (the fact that the trajectories of the geodesic flow on a flat surface is in one-to-one correspondence with the trajectories of a billiard table), the relation with interval exchange transformations, including Rauzy-Veech induction and renormalization, Veech’s zippered rectangles, the study of flat structures arising from holomorphic differential forms, their relation with Teichmüller theory, in particular Teichmüller discs arising from square-tiled surfaces and Veech surfaces.
The paper includes an exposition of recent results of McMullen, Calta and others on the subject. The author motivates the following problem, which he calls the main conjecture: prove a structure theorem that shows that the closure of the complex geodesics (like Teichmüller discs) in moduli space is analogous to the corrresponding situation in a homogeneus space. (It is known that the moduli space is not a homogeneous space.)
There are several other open problems that are mentioned in this paper, concerning the structure of the strata of moduli spaces of flat surfaces, the structure of orbit closures under \(\text{GL}^+(2,\mathbb{R})\) in the moduli space of curves, and other topics.
For the entire collection see [Zbl 1106.11002].

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
30F30 Differentials on Riemann surfaces
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
30F60 Teichmüller theory for Riemann surfaces
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