Dynamical aspects of Teichmüller theory. \(\operatorname{SL}(2,\mathbb R)\)-action on moduli spaces of flat surfaces
This book constitutes a deep survey on different dynamical aspects of Teichmüller theory and the action of \(\mathrm{SL}(2,\mathbb{R})\) on the moduli space of flat surfaces. \par A flat surface is a pair \((X,\omega)\) of a compact Riemann surface \(X\) together with an abelian differential \(\omega \in H^{0}(X,\Omega_{X})\). The moduli space of flat surfaces forms then a bundle over the moduli space of curves and it is stratified according to the multiplicities of the zeroes of the differential. The name flat surface comes from the fact that one can define a flat metric on \(X\) away from the zeroes of \(\omega\) and realise \((X,\omega)\) as a union of polygons with parallel sides identified by translations. This fact allows us to define a natural action of \(\mathrm{SL}(2,\mathbb{R})\) on the strata of the moduli space of flat surfaces by letting the matrices act by affine transformations on the polygons. The dynamics of this action are incredibly interesting from different points of view and have applications to several and very different problems such as the study of trajectories in polygonal billiards or of the Teichmüller geodesic flow on the moduli space of curves. \par The first chapter introduces the different aspects of the theory needed in the rest of the book, including the moduli space of abelian differentials, its stratification and the action of \(\mathrm{SL}(2,\mathbb{R})\) on it, the relation with the Teichmüller flow and the Kontsevich-Zorich cocycle, and the definition and main properties of Teichmüller curves. \par The rest of the chapters in the book are completely independent of each other. Each of them focuses on some of the author's results, which serve as a common thread to present different aspects of the theory. \par In Chapter 2 the author presents the proof of the regularity conjecture of Eskin, Kontsevich and Zorich, shown by him and his coauthors in [\textit{A. Avila}, et al., Geom. Funct. Anal. 23, No. 6, 1705--1729 (2013; Zbl 1316.37016)]. This result states that all \(\mathrm{SL}(2,\mathbb{R})\)-invariant measures in moduli spaces of flat surfaces satisfy certain regularity condition. This condition is a key ingredient needed in the proof of the famous formula of Eskin-Kontsevich-Zorich relating the sum of non-negative Lyapunov exponents to Siegel-Veech constants. \par Chapter 3 focuses on the mixing property of ergodic \(\mathrm{SL}(2,\mathbb{R})\)-invariant measures on connected components of strata of abelian differentials and on the study of their rate of mixing. More precisely, the main objective of the chapter is the construction of an infinite family of arithmetic Teichmüller curves in increasing genus, for which the exponential rate of mixing of the associated measures tends to zero. This result, proved by the author and \textit{G. Weitze-Schmithüsen} in [Bull. Soc. Math. France 141, No. 4, 557--602 (2013; Zbl 1356.37051)], shows that there is no uniform lower bound to the exponential rate of mixing as one varies through all strata. \par Chapter 4 deals with algebraically primitive Teichmüller curves, that is closed orbits of the action of \(\mathrm{SL}(2,\mathbb{R})\) on the space of area-1 flat surfaces of genus \(g\), whose trace field achieves the maximum possible degree \(g\) over \(\mathbb{Q}\). Non-arithmetic Teichmüller curves, that is those not arising from branched covers of tori, tend to be very rare in each genus and, in fact, several finiteness results have been proved in the last years. The particular results presented in this chapter, which were originally published in [the author and \textit{A. Wright}, Duke Math. J. 164, No. 6, 1041--1077 (2015; Zbl 1345.37029)], show the finiteness of algebraically primitive Teichmüller curves in the minimal stratum for prime genus \(g\ge 3\). \par Chapter 5 studies the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle on the moduli space of flat surfaces in genus \(g\). The Lyapunov exponents are \(2g\) real numbers \(\lambda_{1}\ge\dots\ge\lambda_{2g}\) encoding the growth rate of the norm of cohomology classes through parallel transport along the geodesic flow. The Lyapunov spectrum is called simple if all exponents appear with multiplicity one, that is if the above inequalities are strict. The first result of the chapter, which appeared originally in [\textit{A. Eskin} and the author, Geom. Dedicata 179, 45--67 (2015; Zbl 1357.37055)], shows that Lyapunov exponents on Teichmüller curves can be computed via random products of matrices. The rest of the chapter is devoted to the explanation of two associated results: a criterion for the simplicity of the Lyapunov spectrum of arithmetic Teichmüller curves based on Galois theory, from [the author et al., Invent. Math. 202, No. 1, 333--425 (2015; Zbl 1364.37081)], and a counterexample to a conjecture of Forni relating the non-vanishing of the Lyapunov exponents with the existence of certain cylinder decomposition, from [\textit{V. Delecroix} and the author, Math. Res. Lett. 22, No. 6, 1667--1678 (2015; Zbl 1360.37087)]. \par Finally, Chapter 6 deals with the monodromies of the Kontsevich-Zorich cocycle and Filip's classification of such monodromy groups. This classification establishes all the possible groups, but does not determine which of them are actually realisable in the context of the Kontsevich-Zorich cocycle. In a joint work with \textit{S. Filip} and \textit{G. Forni} [J. Eur. Math. Soc. 20, No. 1, 165--198 (2018; Zbl 1388.37030)], the author proved the realisability of one of the groups in the list by constructing an arithmetic Teichmüller curve for which the monodromy group of the Kontsevich-Zorich cocycle is \(\operatorname{SO}^{*}(6)\).
scientific article; zbMATH DE number 6897814