Posted by: matheuscmss | December 9, 2016

“Quelques contributions à la théorie de l’action de SL(2,R) sur les espaces de modules de surfaces plates”

I have just upload to the arXiv the memoir of my `Habilitation à Diriger des Recherches’ (HDR) dossier.

This memoir is a preliminary step towards obtaining a HDR diploma (a general requirement in the French academic system to supervise PhD students, etc.) and it summarizes some of my researches after my PhD thesis (or, more specifically, my researches on the dynamics of the Teichmüller flow).

Despite the title and abstract in French, the main part of this memoir is in English.

The first chapter of the memoir recalls many basic facts on the SL(2,\mathbb{R}) action on the moduli spaces of translation surfaces. In particular, this chapter is a general introduction to all subsequent chapters of the memoir. Here, the exposition is inspired from Zorich’s survey, Yoccoz’s survey and our survey with Forni.

After reading the first chapter, the reader is free to decide the order in which the remainder of the memoir will be read: indeed, the subsequent chapters are independent from each other.

The second chapter of the memoir is dedicated to the main result in my paper with Avila and Yoccoz on the Eskin-Kontsevich-Zorich regularity conjecture. Of course, the content of this chapter borrows a lot from my four blog posts on this subject.

The third chapter of the memoir discusses my paper with Schmithüsen on complementary series (and small spectral gap) for explicit families of arithmetic Teichmüller curves (i.e., SL(2,\mathbb{R})-orbits of square-tiled surfaces).

The fourth chapter of the memoir is dedicated to my paper with Wright on the applications of the notion of Hodge-Teichmüller planes to the question of classification of algebraically primitive Teichmüller curves. Evidently, some portions of this chapter are inspired by my blog posts on this topic.

The fifth chapter of the memoir is consacrated to Lyapunov exponents of the Kontsevich-Zorich cocycle over arithmetic Teichmüller curves. Indeed, after explaining the results in my paper with Eskin about the applications of Furstenberg boundaries to the simplicity of Lyapunov exponents, we spend a large portion of the chapter discussing the Galois-theoretical criterion for simplicity of Lyapunov exponents developed in my paper with Möller and Yoccoz. Finally, we conclude this chapter with an application of these results (obtained together with Delecroix in this paper here) to a counter-example to a conjecture of Forni.

The last chapter of the memoir is based on my paper with Filip and Forni on the construction of examples of `exotic’ Kontsevich-Zorich monodromy groups and it is essentially a slightly modified version of this blog post here.

Closing this short post, let me notice that the current version of this memoir still has no acknowledgements (except for a dedicatory to Jean-Christophe Yoccoz’s memory) because I plan to add them only after I get the referee reports. Logically, once I get the feedback from the referees, I’m surely going to include acknowledgements to my friends/coauthors who made this memoir possible! 😀

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