Teichmüller space

About: Teichmüller space is a research topic. Over the lifetime, 1283 publications have been published within this topic receiving 26716 citations.

A Primer on Mapping Class Groups (Pms-49)

Abstract: The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmuller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Geometry and Spectra of Compact Riemann Surfaces

Abstract: Preface.-Chapter 1: Hyperbolic Structures.-Chapter 2: Trigonometry.- Chapter 3: Y-Pieces and Twist Parameters.- Chapter 4:The Collar Theorem.- Chapter 5: Bers' Constant and the Hairy Torus.- Chapter 6: The Teichmuller Space.- Chapter 7: The Spectrum of the Laplacian.- Chapter 8: Small Eigenvalues.- Chapter 9: Closed Geodesics and Huber's Theorem.- Chapter 10: Wolpert's Theorem.- Chapter 11: Sunada's Theorem.- Chapter 12: Examples of Isospectral Riemann surfaces.- Chapter 13: The Size of Isospectral Families.- Chapter 14: Perturbations of the Laplacian in Hilbert Space.-Appendix: Curves and Isotopies.-Bibliography.-Index.-Glossary.

Geometry of the complex of curves I: Hyperbolicity

Abstract: The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.

The decorated Teichmüller space of punctured surfaces

Abstract: A principal ℝ + 5 -bundle over the usual Teichmuller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmuller space,” which we hope will provide a formalism for understanding some connections between the Teichmuller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.

The virtual cohomological dimension of the mapping class group of an orientable surface

Abstract: Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those of the arithmetic groups. If G is a linear algebraic group defined over Q, X is the symmetric space associated to G and A is an arithmetic subgroup of G, then A is virtually torsion free and acts properly discontinuously on X. The rational homology of A is the same as that of X/A. Furthermore there is a "bordification" of X ([BS]) to a manifold with corners 3~ and an extension of the action of A to a properly discontinuous action on Jf so that the quotient X / A is compact. The boundary of Jf is homotopy equivalent to a wedge of spheres, say of dimension d, and the virtual cohomological dimension of A is n d + 1, where n is the dimension of X. In the case of the mapping class group there is no analog for G, Ivanov (unpublished) has proven that F is not arithmetic. Nevertheless, F acts properly discontinuously on Teichmiiller space r which is homeomorphic to Euclidean space (of dimension 6 g 6 + 2 s ) ; 3"will play the role of the symmetric space. The quotient of .9by F is the moduli space of curves whose rational homology is then identified with that of E Harvey [Har] has constructed a Borel-Serre bordification J of 3-by analytic methods. In the case where F has punctures, we will build g by a different, combinatorial method. In addition, we will explicitly describe inside Y a cell complex Y of dimension 4 g 4 + s onto which J may be F-equivariantly retracted, thus establishing an analog of the constructions for SL n by Serre, Soul6 ([Sol) and Ash ([A]). This complex will be of the lowest possible dimension because we will use J= to prove our main result: