Graduate Texts in Mathematics 243 - maths.ed.ac.uk

1 Graduate Texts in MathematicsEditorial BoardS. Ribet243 Graduate Texts in Mathematics1 TAKEUTI/ZARING. Introduction to AxiomaticSet Theory. 2nd Measure and Category. 2nd Topological Vector A Course inHomological Algebra. 2nd Categories for the WorkingMathematician. 2nd Projective Axiomatic Set Introduction to Lie Algebras andRepresentation COHEN. A Course in Simple CONWAY. Functions of One Complex VariableI. 2nd BEALS. Advanced Mathematical ANDERSON/FULLER. Rings and Categories ofModules. 2nd GOLUBITSKY/GUILLEMIN. Stable Mappings andTheir BERBERIAN. Lectures in Functional Analysisand Operator WINTER. The Structure of ROSENBLATT. Random Processes. 2nd HALMOS. Measure HALMOS. A Hilbert Space Problem HUSEMOLLER. Fibre Bundles. 3rd HUMPHREYS. Linear Algebraic BARNES/MACK. An Algebraic Introduction toMathematical GREUB.

2 Linear Algebra. 4th HOLMES. Geometric Functional Analysis andIts HEWITT/STROMBERG. Real and MANES. Algebraic KELLEY. General ZARISKI/SAMUEL. Commutative l . I .29 ZARISKI/SAMUEL. Commutative JACOBSON. Lectures in Abstract Algebra JACOBSON. Lectures in Abstract Algebra JACOBSON. Lectures in Abstract Algebra of Fields and Galois HIRSCH. Differential SPITZER. Principles of Random Walk. 2nd ALEXANDER/WERMER. Several ComplexVariables and Banach Algebras. 3rd KELLEY/NAMIOKAet al. Linear MONK. Mathematical GRAUERT/FRITZSCHE. Several ComplexVa r i a b l e s .39 ARV E S O N. An Invitation KEMENY/SNELL/KNAPP. Denumerable MarkovChains. 2nd APOSTOL. Modular Functions and DirichletSeries in Number Theory. 2nd SERRE. Linear Representations of GILLMAN/JERISON. Rings of KENDIG.

3 Elementary Algebraic MOISE. Geometric Topology in Dimensions 2and SACHS/WU. General Relativity GRUENBERG/WEIR. Linear Geometry. 2nd EDWARDS. Fermat s Last KLINGENBERG. A Course in HARTSHORNE. Algebraic MANIN. A Course in Mathematical GRAVER/WATKINS. Combinatorics withEmphasis on the Theory of BROWN/PEARCY. Introduction to OperatorTheory I: Elements of Functional MASSEY. Algebraic Topology: CROWELL/FOX. Introduction to Knot Numbers,p-adic Analysis,and Zeta-Functions. 2nd LANG. Cyclotomic ARNOLD. Mathematical Methods in ClassicalMechanics. 2nd WHITEHEAD. Elements of Homotopy KARGAPOLOV/MERIZJAKOV. Fundamentals ofthe Theory of BOLLOBAS. Graph EDWARDS. Fourier Series. Vol. I. 2nd WELLS. Differential Analysis on ComplexManifolds. 2nd WATERHOUSE. Introduction to Affine SERRE.

4 Local Operators in LANG. Cyclotomic Fields MASSEY. Singular Homology FARKAS/KRA. Riemann Surfaces. 2nd STILLWELL. Classical Topology andCombinatorial Group Theory. 2nd HUNGERFORD. DAV E N P O RT. Multiplicative Number HOCHSCHILD. Basic Theory of AlgebraicGroups and Lie Algebras.(continued after index) A Course in VE. Probability Theory I. 4th VE. Probability Theory II. 4th Geoghegan Topological Methods in GroupTheoryEditorial BoardS. RibetMathematics DepartmentMathematics DepartmentUniversity of California at on acid-free of Mathematical University (SUNY)San Francisco, CA 94132 BinghamtonSan Francisco State Subject Classification (2000): 20-xx 54xx 57-xx 53-xxAll rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.

5 Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 2008 Springer Science+Business Media, LLCor by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this e-ISBN 978-0-387-74714-2 Ross GeogheganISBN 978-0-387-74611-1 Library of Congress Control Number: 2007940952NY 13902-6000 Berkeley, CA 94720-3840To Suzanne, Niall and MichaelPrefaceThis book is about the interplay between algebraic topology and the theoryof infinite discrete groups. I have written it for three kinds of readers. First,it is for Graduate students who have had an introductory course in algebraictopology and who need bridges from common knowledge to the current re-search literature in geometric and homological group theory.

6 Secondly, I amwriting for group theorists who would like to know more about the topologicalside of their subject but who have been too long away from topology. Thirdly,I hope the book will be useful to manifold topologists, both high- and low-dimensional, as a reference source for basic material on proper homotopy andlocally finite keep the length reasonable and the focus clear, I assume that the readerknows or can easily learn the necessary algebra, but wants to see the topologydone in detail. Scattered through the book are sections entitled Review of .. in which I give statements, without proofs, of most of the algebraic theoremsused. Occasionally the algebraic references are more conveniently included inthe course of a topological discussion. All of this algebra is standard, and canbe found in many textbooks. It is a mixture of homological algebra, combina-torial group theory, a little category theory, and a little module theory.

7 I for topology, I assume only that the reader has or can easily reacquireknowledge of elementary general topology. Nearly all of what I use is sum-marized in the opening section. A prior course on fundamental group andsingular homology is desirable, but not absolutely essential if the reader iswilling to take a very small number of theorems in Chap. 2 on faith (or, witha different philosophy, as axioms). But this is not an elementary book. Mymaxim has been: Start far back but go fast. In my choice of topological material, I have tried to minimize the overlapwith related books such as [29], [49], [106], [83], [110], [14] and [24]. There issome overlap of technique with [91], mainly in the content of my Chap. 11,but the point of that book is different, as it is pitched towards problems ingeometric book is divided into six Parts. Parts I and III could be the basis for auseful course in algebraic topology (which might also include Sects.)

8 I have divided this material up, and placed it, with group theory in II is about finiteness properties of groups, including both the theoryand some key examples. This is a topic that does not involve asymptotic orend-theoretic invariants. By contrast, Parts IV and V are mostly concernedwith such matters topological invariants of a group which can be seen atinfinity. Part VI consists of essays on three important topics related to, butnot central to, the thrust of the modern study of infinite groups brings several areas of mathematicsinto contact with group theory. Standing out among these are: Riemanniangeometry, synthetic versions of non-positive sectional curvature ( , hyper-bolic groups, CAT(0) spaces), homological algebra, probability theory, coarsegeometry, and topology. My main goal is to help the reader with the last more detail, I distinguish between topological methods (the subject ofthis book) and metric methods.

9 The latter include some topics touched on herein so far as they provide enriching examples ( , quasi-isometric invariants,CAT(0) geometry, hyperbolic groups), and important methods not discussedhere at all ( , train-tracks in the study of individual automorphisms offree groups, as well as, more broadly, the interplay between group theory andthe geometry of surfaces.)Some of these omitted topics are covered in recentbooks such as [48], [134], [127], [5] and [24].I am indebted to many people for encouragement and support during aproject which took far too long to complete. Outstanding among these areCraig Guilbault, Peter Hilton, Tom Klein, John Meier and Michael late Karl Gruenberg suggested that there is a need for this kind of book,and I kept in mind his guidelines. Many others helped as well too many tolist; among those whose suggestions are incorporated in the text are: DavidBenson, Robert Bieri, Matthew Brin, Ken Brown, Kai-Uwe Bux, Dan Far-ley, Wolfgang Kappe, Peter Kropholler, Francisco Fernandez Lasheras, GeraldMarchesi, Holgar Meinert, Boris Okun, Martin Roller, Ralph Strebel, GaddeSwarup, Kevin Whyte, and David have included Source Notes after some of the sections.

10 I would like tomake clear that these constitute merely a subjective choice, mostly paperswhich originally dealt with some of the less well-known topics. Other papersand books are listed in the Source Notes because I judge they would be usefulfor further reading. I have made no attempt to give the kind of bibliographywhich would be appropriate in an authoritative survey. Indeed, I have omittedattribution for material that I consider to be well-known, or folklore, or (andthis applies to quite a few items in the book) ways of looking at things whichemerge naturally from my approach, but which others might consider to be folklore .Lurking in the background throughout this book is what might be calledthe shape-theoretic point of view. This could be summarized as the transferPrefaceIXof the ideas of Borsuk s shape theory of compact metric spaces (later enrichedby the formalism of Grothendieck s pro-categories ) to the proper homotopytheory of ends of open manifolds and locally compact polyhedra, and then, inthe case of universal covers of compact polyhedra, to group theory.