Topological methods in group theory

472 Pages·2008·3.053 MB·English

by  Geoghegan R.

243
Graduate Texts in Mathematics
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S.Axler
K.A.Ribet
Graduate Texts in Mathematics
1 TAKEUTI/ZARING.IntroductiontoAxiomatic 38 GRAUERT/FRITZSCHE.SeveralComplex
SetTheory.2nded. Variables.
2 OXTOBY.MeasureandCategory.2nded. 39 ARVESON.AnInvitationtoC-Algebras.
3 SCHAEFER.TopologicalVectorSpaces. 40 KEMENY/SNELL/KNAPP.DenumerableMarkov
2nded. Chains.2nded.
4 HILTON/STAMMBACH.ACoursein 41 APOSTOL.ModularFunctionsandDirichlet
HomologicalAlgebra.2nded. SeriesinNumberTheory.2nded.
5 MACLANE.CategoriesfortheWorking 42 J.-P.SERRE.LinearRepresentationsofFinite
Mathematician.2nded. Groups.
6 HUGHES/PIPER.ProjectivePlanes. 43 GILLMAN/JERISON.RingsofContinuous
7 J.-P.SERRE.ACourseinArithmetic. Functions.
8 TAKEUTI/ZARING.AxiomaticSetTheory. 44 KENDIG.ElementaryAlgebraicGeometry.
9 HUMPHREYS.IntroductiontoLieAlgebrasand 45 LOÈVE.ProbabilityTheoryI.4thed.
RepresentationTheory. 46 LOÈVE.ProbabilityTheoryII.4thed.
10 COHEN.ACourseinSimpleHomotopy 47 MOISE.GeometricTopologyinDimensions2
Theory. and3.
11 CONWAY.FunctionsofOneComplexVariable 48 SACHS/WU.GeneralRelativityfor
I.2nded. Mathematicians.
12 BEALS.AdvancedMathematicalAnalysis. 49 GRUENBERG/WEIR.LinearGeometry.2nded.
13 ANDERSON/FULLER.RingsandCategoriesof 50 EDWARDS.Fermat’sLastTheorem.
Modules.2nded. 51 KLINGENBERG.ACourseinDifferential
14 GOLUBITSKY/GUILLEMIN.StableMappingsand Geometry.
TheirSingularities. 52 HARTSHORNE.AlgebraicGeometry.
15 BERBERIAN.LecturesinFunctionalAnalysis 53 MANIN.ACourseinMathematicalLogic.
andOperatorTheory. 54 GRAVER/WATKINS.Combinatoricswith
16 WINTER.TheStructureofFields. EmphasisontheTheoryofGraphs.
17 ROSENBLATT.RandomProcesses.2nded. 55 BROWN/PEARCY.IntroductiontoOperator
18 HALMOS.MeasureTheory. TheoryI:ElementsofFunctionalAnalysis.
19 HALMOS.AHilbertSpaceProblemBook. 56 MASSEY.AlgebraicTopology:An
2nded. Introduction.
20 HUSEMOLLER.FibreBundles.3rded. 57 CROWELL/FOX.IntroductiontoKnotTheory.
21 HUMPHREYS.LinearAlgebraicGroups. 58 KOBLITZ.p-adicNumbers,p-adicAnalysis,
22 BARNES/MACK.AnAlgebraicIntroductionto andZeta-Functions.2nded.
MathematicalLogic. 59 LANG.CyclotomicFields.
23 GREUB.LinearAlgebra.4thed. 60 ARNOLD.MathematicalMethodsinClassical
24 HOLMES.GeometricFunctionalAnalysisand Mechanics.2nded.
ItsApplications. 61 WHITEHEAD.ElementsofHomotopyTheory.
25 HEWITT/STROMBERG.RealandAbstract 62 KARGAPOLOV/MERIZJAKOV.Fundamentalsof
Analysis. theTheoryofGroups.
26 MANES.AlgebraicTheories. 63 BOLLOBAS.GraphTheory.
27 KELLEY.GeneralTopology. 64 EDWARDS.FourierSeries.Vol.I.2nded.
28 ZARISKI/SAMUEL.CommutativeAlgebra. 65 WELLS.DifferentialAnalysisonComplex
Vol.I. Manifolds.2nded.
29 ZARISKI/SAMUEL.CommutativeAlgebra. 66 WATERHOUSE.IntroductiontoAffineGroup
Vol.II. Schemes.
30 JACOBSON.LecturesinAbstractAlgebraI. 67 SERRE.LocalFields.
BasicConcepts. 68 WEIDMANN.LinearOperatorsinHilbert
31 JACOBSON.LecturesinAbstractAlgebraII. Spaces.
LinearAlgebra. 69 LANG.CyclotomicFieldsII.
32 JACOBSON.LecturesinAbstractAlgebraIII. 70 MASSEY.SingularHomologyTheory.
TheoryofFieldsandGaloisTheory. 71 FARKAS/KRA.RiemannSurfaces.2nded.
33 HIRSCH.DifferentialTopology. 72 STILLWELL.ClassicalTopologyand
34 SPITZER.PrinciplesofRandomWalk.2nded. CombinatorialGroupTheory.2nded.
35 ALEXANDER/WERMER.SeveralComplex 73 HUNGERFORD.Algebra.
VariablesandBanachAlgebras.3rded. 74 DAVENPORT.MultiplicativeNumberTheory.
36 KELLEY/NAMIOKAetal.LinearTopological 3rded.
Spaces. 75 HOCHSCHILD.BasicTheoryofAlgebraic
37 MONK.MathematicalLogic. GroupsandLieAlgebras.
(continuedafterindex)
Ross Geoghegan
Topological Methods in Group
Theory
Ross Geoghegan
Department of Mathematical Sciences
Binghamton University (SUNY)
Binghamton
NY 13902-6000
USA
[email protected]
EditorialBoard
S.Axler K.A.Ribet
MathematicsDepartment MathematicsDepartment
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
[email protected] [email protected]
ISBN978-0-387-74611-1 e-ISBN978-0-387-74714-2
LibraryofCongressControlNumber:2007940952
MathematicsSubjectClassification(2000):20-xx 54xx 57-xx 53-xx
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
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To Suzanne, Niall and Michael
Preface
This book is about the interplay between algebraic topology and the theory
of 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 algebraic
topology and who need bridges from common knowledge to the current re-
search literature in geometric and homological group theory. Secondly, I am
writingforgrouptheoristswhowouldliketoknowmoreaboutthetopological
sideoftheirsubjectbutwhohavebeentoolongawayfromtopology.Thirdly,
I hope the book will be useful to manifold topologists, both high- and low-
dimensional,as areferencesourceforbasic materialonproperhomotopyand
locally finite homology.
Tokeepthelengthreasonableandthefocusclear,Iassumethatthereader
knowsorcaneasilylearnthenecessaryalgebra,butwantstoseethetopology
doneindetail.Scatteredthroughthebookaresectionsentitled“Reviewof...”
in which I give statements, without proofs, of most of the algebraictheorems
used. Occasionally the algebraic references are more conveniently included in
the courseof a topologicaldiscussion.All ofthis algebrais standard,andcan
be foundinmanytextbooks.Itisa mixtureofhomologicalalgebra,combina-
torialgrouptheory,a little categorytheory,anda little module theory.I give
references.
As for topology, I assume only that the reader has or can easily reacquire
knowledge of elementary general topology. Nearly all of what I use is sum-
marized in the opening section. A prior course on fundamental group and
singular homology is desirable, but not absolutely essential if the reader is
willing to take a very small number of theorems in Chap. 2 on faith (or,with
a different philosophy, as axioms). But this is not an elementary book. My
maxim has been: “Start far back but go fast.”
In my choice of topological material, I have tried to minimize the overlap
with related books such as [29], [49], [106], [83], [110], [14] and [24]. There is
some 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 in
geometric topology.
VIII Preface
The book is dividedinto six Parts.PartsIandIII couldbe the basisfor a
usefulcourseinalgebraictopology(whichmightalsoincludeSects.16.1-16.4).
I have divided this material up, and placed it, with group theory in mind.
Part II is about finiteness properties of groups, including both the theory
and some key examples. This is a topic that does not involve asymptotic or
end-theoretic invariants. By contrast, Parts IV and V are mostly concerned
with such matters – topological invariants of a group which can be seen “at
infinity.” Part VI consists of essays on three important topics related to, but
not central to, the thrust of the book.
The modern study of infinite groups brings several areas of mathematics
into contact with group theory. Standing out among these are: Riemannian
geometry, synthetic versions of non-positive sectional curvature (e.g., hyper-
bolic groups,CAT(0) spaces),homologicalalgebra,probabilitytheory,coarse
geometry, and topology. My main goal is to help the reader with the last of
these.
In more detail, I distinguish between topological methods (the subject of
thisbook)andmetricmethods.Thelatterincludesometopicstouchedonhere
in so far as they provide enriching examples (e.g., quasi-isometric invariants,
CAT(0) geometry, hyperbolic groups), and important methods not discussed
here at all (e.g., train-tracks in the study of individual automorphisms of
free groups,aswell as,morebroadly,the interplaybetween grouptheory and
the geometry of surfaces.) Some of these omitted topics are coveredin recent
books such as [48], [134], [127], [5] and [24].
I am indebted to many people for encouragement and support during a
project which took far too long to complete. Outstanding among these are
Craig Guilbault, Peter Hilton, Tom Klein, John Meier and Michael Mihalik.
The late KarlGruenbergsuggestedthat there is a needfor this kindof book,
and I kept in mind his guidelines. Many others helped as well – too many to
list; among those whose suggestions are incorporated in the text are: David
Benson, Robert Bieri, Matthew Brin, Ken Brown, Kai-Uwe Bux, Dan Far-
ley,WolfgangKappe,PeterKropholler,FranciscoFernandezLasheras,Gerald
Marchesi, Holgar Meinert, Boris Okun, Martin Roller, Ralph Strebel, Gadde
Swarup, Kevin Whyte, and David Wright.
I have included Source Notes after some of the sections. I would like to
make clear that these constitute merely a subjective choice, mostly papers
which originally dealt with some of the less well-known topics. Other papers
andbooksarelistedintheSourceNotesbecauseIjudgethey wouldbeuseful
for further reading. I have made no attempt to give the kind of bibliography
whichwouldbeappropriateinanauthoritativesurvey.Indeed,Ihaveomitted
attributionformaterialthatIconsidertobewell-known,or“folklore,”or(and
this applies to quite a few items in the book) ways oflooking at things which
emerge naturally from my approach, but which others might consider to be
“folklore”.
Lurking in the background throughout this book is what might be called
the“shape-theoreticpointofview.”Thiscouldbesummarizedasthetransfer
Preface IX
oftheideasofBorsuk’sshapetheoryofcompactmetricspaces(laterenriched
bytheformalismofGrothendieck’s“pro-categories”)totheproperhomotopy
theoryofends ofopenmanifoldsandlocallycompactpolyhedra,andthen,in
thecaseofuniversalcoversofcompactpolyhedra,togrouptheory.Ioriginally
set out this program, in a sense the outline of this book, in [68]. The forma-
tive ideas for this developed as a result of extensive conversations with, and
collaboration with, David A. Edwards in my mathematical youth. Though
those conversations did not involve group theory, in some sense this book is
an outgrowth of them, and I am happy to acknowledge his influence.
SpringereditorMarkSpencerwaseversupportive,especiallywhenImade
the decision, at a late stage, to reorganize the book into more and shorter
chapters (eighteen instead of seven). Comments by the anonymous referees
were also helpful.
IhadthebenefitoftheTeXexpertiseofMargePratt;besideshereverpa-
tient and thoughtful consideration, she typed the book superbly. I am also
grateful for technical assistance given me by my mathematical colleagues
Collin Bleak, Keith Jones and Erik K. Pedersen, and by Frank Ganz and
Felix Portnoy at Springer.
Finally,theencouragementtofinishgivenmebymywifeSuzanneandmy
sons Niall and Michael was a spur which in the end I could not resist.
Binghamton University (SUNY Binghamton),
May 2007
Ross Geoghegan
X Preface
Notes to the Reader
1. Shorter courses: Within this book there are two natural shorter courses.
Bothbeginwith the firstfour sections ofChap.1onthe elementarytopology
of CW complexes. Then one can proceed in either of two ways:
• The homotopical course: Chaps. 3, 4, 5 (omitting 5.4), 6, 7, 9, 10, 16 and
17.
• The homological course: Chaps. 2, 5, 8, 11, 12, 13, 14 and 15.
2. Notation: If the group G acts on the space X on the left, the set of orbits
is denoted by G\X;similarly,a rightactiongivesX/G.But if R is a commu-
tative ring and (M,N) is a pair of R-modules, I always write M/N for the
quotientmodule. And ifA is a subspaceof the space X,the quotientspace is
denoted by X/A.
The term “ring” without further qualification means a commutative ring
with 1(cid:1)=0.
I draw attention to the notation X −c A where A is a subcomplex of the
CW complex X. This is the “CW complement”, namely the largest subcom-
plex of X whose 0-skeleton consists of the vertices of X which are not in
A. If one wants to stay in the world of CW complexes one must use this as
“complement”sinceingeneralthe ordinarycomplementis notasubcomplex.
The notations A := B and B =: A both mean that A (a new symbol) is
defined to be equal to B (something already known).
As usual, the non-word“iff” is short for “if and only if.”
3. Categories: I assume an elementary knowledge of categories and func-
tors. I sometimes refer to well-known categories by their objects (the word is
given a capital opening letter). Thus Groups refers to the category of groups
and homomorphisms. Similarly: Sets, Pointed Sets, Spaces, Pointed Spaces,
Homotopy (spaces and homotopy classes of maps), Pointed Homotopy, and
R-modules.When there mightbe ambiguity Iname the objects andthe mor-
phisms(e.g.,the categoryoforientedCWcomplexesoflocallyfinitetypeand
CW-proper homotopy classes of CW-proper maps).
4. Website: I plan to collect corrections,updates etc. at the Internet website
math.binghamton.edu/ross/tmgt
I also invite supplementary essays or comments which readers feel would
be helpful, especially to students. Such contributions, as well as corrections
and errata, should be sent to me at the web address
[email protected]
Contents
PART I: ALGEBRAIC TOPOLOGY FOR GROUP THEORY 1
1 CW Complexes and Homotopy ............................. 3
1.1 Review of general topology ............................... 3
1.2 CW complexes .......................................... 10
1.3 Homotopy .............................................. 23
1.4 Maps between CW complexes............................. 28
1.5 Neighborhoods and complements .......................... 31
2 Cellular Homology ......................................... 35
2.1 Review of chain complexes................................ 35
2.2 Review of singular homology.............................. 37
2.3 Cellular homology: the abstract theory ..................... 40
2.4 The degree of a map from a sphere to itself ................. 43
2.5 Orientation and incidence number ......................... 52
2.6 The geometric cellular chain complex ...................... 60
2.7 Some properties of cellular homology....................... 62
2.8 Further properties of cellular homology..................... 65
2.9 Reduced homology ...................................... 70
3 Fundamental Group and Tietze Transformations ........... 73
3.1 Fundamental group, Tietze transformations, Van Kampen
Theorem ............................................... 73
3.2 Combinatorial description of covering spaces ................ 84
3.3 Review of the topologically defined fundamental group ....... 94
3.4 Equivalence of the two definitions ......................... 96
4 Some Techniques in Homotopy Theory .....................101
4.1 Altering a CW complex within its homotopy type ...........101
4.2 Cell trading.............................................110
4.3 Domination, mapping tori, and mapping telescopes ..........112