Handbook of Teichmüller Theory, Volume IV, 2014

2007

We study the boundary of Teichmüller disks in T g , a partial compactification of Teichmüller space, and their image in Schottky space. We give a broad introduction to Teichmüller disks and explain the relation between Teichmüller curves and Veech groups. Furthermore, we describe Braungardt's construction of T g and compare it with the Abikoff augmented Teichmüller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary of T g. This prepares the description of boundary points that a Teichmüller disk has, with a particular emphasis to the case that it leads to a Teichmüller curve. Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmüller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmüller disks in Schottky space and show that one can choose the projection from Teichmüller space to Schottky space in such a manner that the image of the Teichmüller disk is a quotient by an infinite group.

Mathematical Surveys and Monographs, 1999

We present an outline of the theory of universal Teichmüller space, viewed as part of the theory of QS, the space of quasisymmetric homeomorphisms of a circle. Although elements of QS act in one dimension, most results about QS depend on a two-dimensional proof. QS has a manifold structure modelled on a Banach space, and after factorization by P SL(2, R) it becomes a complex manifold. In applications, QS is seen to contain many deformation spaces for dynamical systems acting in one, two and three dimensions; it also contains deformation spaces of every hyperbolic Riemann surface, and in this naive sense it is universal. The deformation spaces are complex submanifolds and often have certain universal properties themselves, but those properties are not the object of this paper. Instead we focus on the analytic foundations of the theory necessary for applications to dynamical systems and rigidity.

Bulletin of the American Mathematical Society

Geometric Function Theory in Higher Dimension, 2017

Advances in Mathematics, 2015

Handbook of Teichmüller Theory, Volume II, 2009

Teichmüller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different fields. Among the fields related to Teichmüller theory, one can surely mention complex analysis, hyperbolic geometry, the theory of discrete groups, algebraic geometry, low-dimensional topology, differential geometry, Lie group theory, symplectic geometry, dynamical systems, number theory, topological quantum field theory, string theory, and there are many others. Let us start by recalling a few definitions. Let S g,p be a connected orientable topological surface of genus g ≥ 0 with p ≥ 0 punctures. Any such surface admits a complex structure, that is, an atlas of charts with values in the complex plane C and whose coordinate changes are holomorphic. In the classical theory, one considers complex structures S g,p for which each puncture of S g,p has a neighborhood which is holomorphically equivalent to a punctured disk in C. To simplify the exposition, we shall suppose that the orientation induced on S g,p by the complex structure coincides with the orientation of this surface. Homeomorphisms of the surface act in a natural manner on atlases, and two complex structures on S g,p are said to be equivalent if there exists a homeomorphism of the surface which is homotopic to the identity and which sends one structure to the other. The surface S g,p admits infinitely many non-equivalent complex structures, except if this surface is a sphere with at most three punctures. To say things precisely, we introduce some notation. Let C g,p be the space of all complex structures on S g,p and let Diff + (S g,p) be the group of orientation-preserving diffeomorphisms of S g,p. We consider the action of Diff + (S g,p) by pullback on C g,p. The quotient space M g,p = C g,p /Diff + (S g,p) is called Riemann's moduli space of deformations of complex structures on S g,p. This space was considered by G. F. B. Riemann in his famous paper on Abelian functions, Theorie der Abel'schen Functionen, Crelle's Journal, Band 54 (1857), in which he studied moduli for algebraic curves. In that paper, Riemann stated, without giving a formal proof, that the space of deformations of equivalence classes of conformal structures on a closed orientable surface of genus g ≥ 2 is of complex dimension 3g − 3. The Teichmüller space T g,p of S g,p was introduced in the 1930s by Oswald Teichmüller. It is defined as the quotient of the space C g,p of complex structures by the group Diff + 0 (S g,p) of orientation-preserving diffeomorphisms of S g,p that are isotopic to the identity. The group Diff + 0 (S g,p) is a normal subgroup of Diff + (S g,p), and the quotient group g,p = Diff + (S g,p)/Diff + 0 (S g,p) is called the mapping class group of S g,p (sometimes also called the modular group, or the Teichmüller modular group)

2008

2007

These notes are based on a lecture course by L. Chekhov held at the University of Manchester in May 2006 and February-March 2007. They are divulgative in character, and instead of containing rigorous mathematical proofs, they illustrate statements giving an intuitive insight. We intentionally remove most bibliographic references from the body of the text devoting a special section to the history of the subject at the end.

Oberwolfach Reports, 2014

The program "New Trends in Teichmüller Theory and Mapping Class Groups" brought together people working in various aspects of the field and beyond. The focus was on the recent developments that include higher Teichmüller theory, the relation with three-manifolds, mapping class groups, dynamical aspects of the Weil-Petersson geodesic flow, and the relation with physics. The goal of bringing together researchers in these various areas, including young PhDs, and promoting interaction and collaboration between them was attained.