Symbolic Dynamics and Hyperbolic Groups
Gromov's theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems. This book is an ela
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen Subseries: Institut de Mathematiques, Universite de Strasbourg Adviser:P.A. Meyer
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Michel Coornaert Athanase Papadopoulos
Symbolic Dynamcis and Hyperbolic Groups
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Michel Coornaert Athanase Papadopoulos Institut de Recherche Mathematique Avancee Universite Louis Pasteur et CNRS 7, rue Rene Descartes F-67084 Strasbourg, France
Mathematics Subject Classification (1991): 53C23, 34C35, 54H20, 58F03, 20F30 ISBN 3-540-56499-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56499-3 Springer-Verlag New York Berlin Heidelberg
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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany
Typesetting: Camera ready by author 46/3140-543210 - Printed on acid-free paper
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Table of contents
Introduction
1
Cbapter 1. - A quick review of Gromov hyperbolic spaces. §1. - Hyperbolic metric spaces. §2. - Hyperbolic groups. §3. - The boundary of a hyperbolic space. §4. - The visual metric on the boundary. §5. - Approximation by trees. §6. - Quasi-geodesics and quasi-isornetries. §7. - Classification of isometries. §8. - The polyhedron Pd(X). Bibliography for Chapter 1.
5 6 8
Chapter 2. - Symbolic dynamics. §1. - Bernoulli shifts. §2. - Expansive systems. §3. - Subshifts of finite type. §4. - Systems of finite type and finitely presented systems. §5. - Symbolic dynamics on N and on 7l . §6. - Sofic systems. Notes and comments on Chapter 2. Bibliography for Chapter 2.
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12 13 13 16 17 18 19
20 22
26 29 31 36
40 41
Chapter 3. - The boundary of a hyperbolic space as a finitely presented dynamical system. 43 §1. - The cocycles X is a geodesic ray, then there exists a point in ax such that r(t n) converges to for every sequence (tn) of real numbers 0 such that t n -> 00. We shall write r( 00). In the same manner, every geodesic '"'I: lR -> X defines two distinct points '"'I ( -00) and '"'I ( 00) of ax .
=
We extend the definition of a geodesic polygon, given in §1, by allowing certain vertices of the polygon to belong to ax. An n-sided geodesic polygon IT = [Xl, ..., x n] is therefore given by n points z j , ... , X n E X U ax (the vertices of IT) and n geodesics
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Chapter 1. -
Review of hyperbolic spaces
/1, ... , /n (the sides of II) with /i joining Xi and Xi+l for each i mod n, Let us recall
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