Dynamical aspects of Teichmüller theory. \(\operatorname{SL}(2,\mathbb R)\)-action on moduli spaces of flat surfaces
type: Item
instance of: scholarly article
title: Dynamical aspects of Teichmüller theory. \(\operatorname{SL}(2,\mathbb R)\)-action on moduli spaces of flat surfaces
zbMATH Open document ID: 1409.37002
DOI: 10.1007/978-3-319-92159-4
author: Carlos Matheus Silva Santos
published in: Atlantis Studies in Dynamical Systems
publication date: 2018-06-29T00:00:00Z
review text: 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.…
reviewed by: D. Torres-Teigell
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 6897814
zbMATH Keywords: Teichmüller dynamics
zbMATH Keywords: moduli spaces
zbMATH Keywords: dynamical systems and geometry
zbMATH Keywords: flat surfaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-92159-4
OpenAlex ID: W4250992137
description: scientific article; zbMATH DE number 6897814