Punctured Torus Groups and 2-Bridge Knot Groups (I)
This monograph is Part 1 of a book project intended to give a full account of Jorgensen's theory of punctured torus Kleinian groups and its generalization, with application to knot theory. Although Jorgensen's original work was not published in complete f
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Hirotaka Akiyoshi Makoto Sakuma Masaaki Wada Yasushi Yamashita
Punctured Torus Groups and 2-Bridge Knot Groups I
1909
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1909
Hirotaka Akiyoshi · Makoto Sakuma Masaaki Wada · Yasushi Yamashita
Punctured Torus Groups and 2-Bridge Knot Groups (I)
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Authors Hirotaka Akiyoshi Osaka City University Advanced Mathematical Institute 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 Japan e-mail: [email protected]
Makoto Sakuma
Masaaki Wada Yasushi Yamashita Department of Information and Computer Sciences Nara Women’s University Kita-uoya Nishimachi Nara 630-8506 Japan e-mail: [email protected] [email protected]
Department of Mathematics Hiroshima University 1-3-1, Kagamiyama Higashi-Hiroshima 739-8526 Japan e-mail: [email protected]
Library of Congress Control Number: 2007925679 Mathematics Subject Classification (2000): 57M50, 30F40, 57M25, 20H10 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-71806-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71806-2 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-71807-9 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 12042029
VA41/3100/SPi
543210
Preface
The main purpose of this monograph is to give a full description of Jorgensen’s theory on the space QF of quasifuchsian (once) punctured torus groups with a complete proof. Our method is based on Poincare’s theorem on fundamental polyhedra. This geometric approach enabled us to extend Jorgensen’s theory beyond the quasifuchsian space and apply to knot theory.
1. History By the late 70’s Troels Jorgensen had made a series of detailed studies on the space QF of quasifuchsian (once) punctured torus groups from the view point of their Ford fundamental domains. These studies are summarized in his famous unfinished paper [40]. In it, he gave a complete description of the combinatorial s
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