Gluing construction and Maskit slice

Sara Maloni

ISBN 10: 3659302694 / ISBN 13: 9783659302695

Published by LAP LAMBERT Academic Publishing Jan 2013, 2013

About this Item

Neuware -In this work we mainly deal with Kleinian groups, which are discrete groups of isometries of the hyperbolic 3¿space. In the 1960s, Kleinian groups were studied mainly analytically, but in the 1970s Thurston revolutionised the subject by taking a more topological viewpoint. In 1990s Keen and Series introduced the Pleating Coordinates Theory. Their key idea was to study the deformation spaces of holomorphic families of Kleinian groups via the internal geometry of the associated hyperbolic 3¿manifold. In this book, given a surface of negative Euler characteristic, we endow it with a projective structure, which depends on some complex parameters, using a `plumbing' construction. In particular, the traces of the holonomy image of the curves on S are polynomials in these parameters, and we prove a formula expressing the coefficients of the top terms of these polynomials in terms of the Dehn-Thurston coordinates of the curves. If the representation is free and discrete, then the representation lies on the Maskit slice, and the formula discussed above enables us to find the asymptotic direction of the pleating rays in the Maskit slice as the bending measure tends to zero.Books on Demand GmbH, Überseering 33, 22297 Hamburg 136 pp. Englisch. Seller Inventory # 9783659302695

Bibliographic Details

Title: Gluing construction and Maskit slice

Publisher: LAP LAMBERT Academic Publishing Jan 2013

Publication Date: 2013

Binding: Taschenbuch

Condition: Neu

About this title

Synopsis:

In this work we mainly deal with Kleinian groups, which are discrete groups of isometries of the hyperbolic 3–space. In the 1960s, Kleinian groups were studied mainly analytically, but in the 1970s Thurston revolutionised the subject by taking a more topological viewpoint. In 1990s Keen and Series introduced the Pleating Coordinates Theory. Their key idea was to study the deformation spaces of holomorphic families of Kleinian groups via the internal geometry of the associated hyperbolic 3–manifold. In this book, given a surface of negative Euler characteristic, we endow it with a projective structure, which depends on some complex parameters, using a `plumbing' construction. In particular, the traces of the holonomy image of the curves on S are polynomials in these parameters, and we prove a formula expressing the coefficients of the top terms of these polynomials in terms of the Dehn-Thurston coordinates of the curves. If the representation is free and discrete, then the representation lies on the Maskit slice, and the formula discussed above enables us to find the asymptotic direction of the pleating rays in the Maskit slice as the bending measure tends to zero.

About the Author:

Sara Maloni obtained her PhD in Mathematics at the University of Warwick. She was then a postdoctoral researcher at the Universities of Toulouse and Paris-Sud, and she will be an assistant professor at Brown University. Her research interests lie at the intersection of hyperbolic geometry, (higher) Teichmüller theory and low-dimensional topology.