Groups Languages and Automata 1st Edition Derek F. Holt
Visit to download the full and correct content document: https://textbookfull.com/product/groups-languages-and-automata-1st-edition-derek-f-h olt/
More products digital (pdf, epub, mobi) instant download maybe you interests ...
Notes on formal languages automata computability and complexity Gallier J
https://textbookfull.com/product/notes-on-formal-languagesautomata-computability-and-complexity-gallier-j/
Machine Vision Algorithms in Java Techniques and Implementation Whelan Paul F Molloy Derek
https://textbookfull.com/product/machine-vision-algorithms-injava-techniques-and-implementation-whelan-paul-f-molloy-derek/
Theory of Groups and Symmetries Finite groups Lie groups and Lie Algebras 1st Edition Alexey P. Isaev
https://textbookfull.com/product/theory-of-groups-and-symmetriesfinite-groups-lie-groups-and-lie-algebras-1st-edition-alexey-pisaev/
Automata and Computability: A Programmer’s Perspective
Ganesh Gopalakrishnan
https://textbookfull.com/product/automata-and-computability-aprogrammers-perspective-ganesh-gopalakrishnan/
Cellular Learning Automata: Theory and Applications
Reza Vafashoar
https://textbookfull.com/product/cellular-learning-automatatheory-and-applications-reza-vafashoar/
Cellular Automata and Discrete Complex Systems 20th International Workshop AUTOMATA 2014 Himeji Japan July 7 9 2014 Revised Selected Papers 1st Edition Teijiro Isokawa
https://textbookfull.com/product/cellular-automata-and-discretecomplex-systems-20th-international-workshop-automata-2014-himejijapan-july-7-9-2014-revised-selected-papers-1st-edition-teijiroisokawa/
Theory of Groups and Symmetries: Representations of Groups and Lie Algebras, Applications 1st Edition Alexey P. Isaev
https://textbookfull.com/product/theory-of-groups-and-symmetriesrepresentations-of-groups-and-lie-algebras-applications-1stedition-alexey-p-isaev/
Language and Automata Theory and Applications Shmuel
Tomi Klein
https://textbookfull.com/product/language-and-automata-theoryand-applications-shmuel-tomi-klein/
Cybercrime and Digital Forensics An Introduction
Thomas J. Holt
https://textbookfull.com/product/cybercrime-and-digitalforensics-an-introduction-thomas-j-holt/
ManagingEditor:ProfessorD.Benson, DepartmentofMathematics,UniversityofAberdeen,UK
49AnintroductiontoK-theoryforC*-algebras,M.RØRDAM,F.LARSEN&N.J.LAUSTSEN
50Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER
51Stepsincommutativealgebra:Secondedition,R.Y.SHARP
52FiniteMarkovchainsandalgorithmicapplications,OLLEHAGGSTR ¨ OM
53Theprimenumbertheorem,G.J.O.JAMESON
54Topicsingraphautomorphismsandreconstruction,JOSEFLAURI&RAFFAELE SCAPELLATO
55Elementarynumbertheory,grouptheoryandRamanujangraphs,GIULIANADAVIDOFF, PETERSARNAK&ALAINVALETTE
56Logic,inductionandsets,THOMASFORSTER
57IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALES etal
58Computationalalgebraicgeometry,HALSCHENCK
59Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories,JOACHIMKOCK
60Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON
61AnintroductiontononcommutativeNoetherianrings:Secondedition,K.R.GOODEARL& R.B.WARFIELD,JR
62Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN
63Singularpointsofplanecurves,C.T.C.WALL
64AshortcourseonBanachspacetheory,N.L.CAROTHERS
65ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,DANIEL SIMSON&ANDRZEJSKOWRO ´ NSKI
66Anintroductiontosievemethodsandtheirapplications,ALINACARMENCOJOCARU& M.RAMMURTY
67Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN
68Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC
69LecturesonKahlergeometry,ANDREIMOROIANU
70Dependencelogic,JOUKUV ¨ A ¨ AN ¨ ANEN
71ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON& ANDRZEJSKOWRO ´ NSKI
72ElementsoftherepresentationtheoryofassociativealgebrasIII,DANIELSIMSON& ANDRZEJSKOWRO ´ NSKI
73Groups,graphsandtrees,JOHNMEIER
74RepresentationtheoremsinHardyspaces,JAVADMASHREGHI
75Anintroductiontothetheoryofgraphspectra,DRAGO ˇ SCVETKOVI ´ C,PETER ROWLINSON&SLOBODANSIMI ´ C
76NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS
77Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV& CHRISTOPHERVOLL
78Cliffordalgebras:Anintroduction,D.J.H.GARLING
79IntroductiontocompactRiemannsurfacesanddessinsd’enfants,ERNESTOGIRONDO& GABINOGONZ ´ ALEZ-DIEZ
80TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN
81Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI
82Finitegeometryandcombinatorialapplications,SIMEONBALL
83Thegeometryofcelestialmechanics,HANSJORGGEIGES
84Randomgraphs,geometryandasymptoticstructure,MICHAELKRIVELEVICH etal
85Fourieranalysis:PartI–Theory,ADRIANCONSTANTIN
86Dispersivepartialdifferentialequations,M.BURAKERDOGAN&NIKOLAOSTZIRAKIS
87Riemannsurfacesandalgebraiccurves,R.CAVALIERI&E.MILES
88Groups,languagesandautomata,DEREKF.HOLT,SARAHREES&CLAASE.R ¨ OVER
Groups,LanguagesandAutomata
DEREKF.HOLT
UniversityofWarwick
SARAHREES
UniversityofNewcastleuponTyne
CLAASE.R OVER
NationalUniversityofIreland,Galway
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi-110002,India 79AnsonRoad,#06-04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org Informationonthistitle:www.cambridge.org/9781107152359
DOI:10.1017/9781316588246
c DerekF.Holt,SarahReesandClaasE.R ¨ over2017
Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-107-15235-9Hardback
ISBN978-1-316-60652-0Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain, accurateorappropriate.
2.5Finitestateautomata,regularlanguagesandgrammars52 2.6Pushdownautomata,context-freelanguagesandgrammars
2.7Turingmachines,recursivelyenumerablelanguages andgrammars
2.8Linearlyboundedautomata,context-sensitivelanguagesandgrammars
2.9Turingmachinesanddecidability
3.1Definitionofthewordproblem
3.2VanKampendiagrams
3.3TheDehnfunction
3.4Thewordproblemasaformallanguage
3.5DehnpresentationsandDehnalgorithms
3.6Fillingfunctions
PARTTWOFINITESTATEAUTOMATAANDGROUPS
4Rewritingsystems
4.1Rewritingsystemsinmonoidsandgroups
4.2Theuseof fsa inthereductionprocess
5.1Definitionofautomaticgroups
5.2Propertiesofautomaticgroups
5.3Shortlexandgeodesicstructures
5.4Theconstructionofshortlexautomaticstructures135
5.5Examplesofautomaticgroups
5.6Closureproperties
5.7Thefalsificationbyfellowtravellerproperty
5.8Stronglygeodesicallyautomaticgroups
5.9Generalisationsofautomaticity
6.1Hyperbolicityconditions
6.2Hyperbolicityforgeodesicmetricspacesandgroups153
6.3Thinbigons,biautomaticityanddivergence
6.4HyperbolicgroupshaveDehnpresentations
6.5GroupswithlinearDehnfunctionsarehyperbolic161
6.6Equivalentdefinitionsofhyperbolicity
6.7Quasigeodesics
6.8Furtherpropertiesofhyperbolicgroups
7.1Introduction
7.2Virtuallyabeliangroupsandrelativelyhyperbolicgroups170
7.3Coxetergroups
7.4Garsidegroups176
7.5Groupswithgeodesicslyinginsomesubclassof Reg 182
7.6Conjugacygeodesics182
8Subgroupsandcosetsystems 184
8.1Rationalandquasiconvexsubsetsofgroups184
8.2Automaticcosetsystems189
9Automatagroups 194
9.1Introducingpermutationaltransducers194
9.2Automatagroups199
9.3Groupsoftreeautomorphisms203
9.4Dualautomata210
9.5Freeautomatagroups212
9.6Decisionproblems213
9.7Resolvingfamousproblems215
10.1TheNovikov–Boonetheorem221 10.2Relatedresults226
11Context-freeandone-counterwordproblems 228
11.1Groupswithcontext-freewordproblem228
11.2Groupswithone-counterwordproblem232
12Context-sensitivewordproblems 236
12.1Lakin’sexample237
12.2Somefurtherexamples239
12.3Fillinglength240
12.4Groupswithlinearfillinglength242
13Wordproblemsinotherlanguageclasses 249
13.1Real-timewordproblems249
13.2Indexedwordproblems254
13.3Poly-context-freewordproblems254
14Theco-wordproblemandtheconjugacyproblem 256
14.1Theco-wordproblem256
14.2Co-context-freegroups257
14.3Indexedco-wordproblems267
14.4Theconjugacyproblem268
Preface
Thisbookexploresconnectionsbetweengrouptheoryandautomatatheory. Weweremotivatedtowriteitbyourobservationsofagreatdiversityofsuch connections;weseeautomatausedtoencodecomplexity,torecogniseaspects ofunderlyinggeometry,toprovideefficientalgorithmsforpracticalcomputation,andmore.
Thebookispitchedatbeginninggraduatestudents,andatprofessionalacademicmathematicianswhoarenotfamiliarwithallaspectsoftheseinterconnectedfields.Itprovidesbackgroundinautomatatheorysufficientforitsapplicationstogrouptheory,andthengivesup-to-dateaccountsofthesevarious applications.Weassumethatthereaderalreadyhasabasicknowledgeofgroup theory,asprovidedinastandardundergraduatecourse,butwedonotassume anypreviousknowledgeofautomatatheory.
Thegroupsthatweconsiderareallfinitelygenerated.Anelementofagroup G isrepresentedasaproductofpowersofelementsofthegeneratingset X , andhenceasastringofsymbolsfrom A := X ∪ X 1,alsocalledwords.Many differentstringsmayrepresentthesameelement.Thegroupmaybedefinedby apresentation;thatis,byitsgeneratingset X togetherwithaset R ofrelations, fromwhichallequationsinthegroupbetweenstringscanbederived.Alternatively,asforinstanceinthecaseofautomatagroups, G mightbedefinedasa groupoffunctionsgeneratedbytheelementsof X . Certainsetsofstrings,alsocalledlanguages,over A arenaturallyofinterest. Westudythewordproblemofthegroup G ,namelythesetWP(G , A)ofstrings over A thatrepresenttheidentityelement.Wedefinealanguagefor G tobea languageover A thatmapsonto G ,andconsiderthelanguageofallgeodesics, andvariouslanguagesthatmapbijectivelyto G .Wealsoconsidercombings, definedtobegrouplanguagesforwhichtwowordsrepresentingeitherthe sameelementorelementsthatareadjacentintheCayleygraphfellowtravel; thatis,theyareataboundeddistantapartthroughouttheirlength.
Preface
Weconsideranautomatontobeadevicefordefininga,typicallyinfinite,set L ofstringsoverafinitealphabet,calledthelanguageoftheautomaton.Any stringoverthefinitealphabetmaybeinputtotheautomaton,andistheneither acceptedifitisin L,orrejectedifitisnot.Weconsiderautomataofvarying degreesofcomplexity,rangingfromfinitestateautomata,whichdefineregular languages,throughpushdownautomata,definingcontext-freelanguages,to Turingmachines,whichsettheboundariesforalgorithmicrecognitionofa language.Inotherwords,weconsiderthefullChomskyhierarchyofformal languagesandtheassociatedmodelsofcomputation.
FinitestateautomatawereusedbyThurstoninhisdefinitionofautomatic groupsafterherealisedthatboththefellowtravellerpropertyandthefinitenessofthesetofconetypesthatCannonhadidentifiedinthefundamental groupsofcompacthyperbolicmanifoldscouldbeexpressedintermsofregularlanguages.Forautomaticgroupsaregularcombingcanbefound;useof thefinitestateautomatonthatdefinesthistogetherwithotherautomatathat encodefellowtravellingallowsinparticularaquadratictimesolutiontothe wordproblem.Word-hyperbolicgroups,asdefinedbyGromov,canbecharacterisedbytheirpossessionofautomaticstructuresofaparticulartype,leading tolineartimesolutionstothewordproblem.
Forsomegroupsthesetofallgeodesicwordsover A isaregularlanguage. Thisistrueforword-hyperbolicgroupsandabeliangroups,withrespecttoany generatingset,andformanyothergroups,includingCoxetergroups,virtually freegroupsandGarsidegroups,forcertaingeneratingsets.Manyofthese groupsareinfactautomatic.
ThepositionofalanguageintheChomskyhierarchycanbeusedasameasureofitscomplexity.Forexample,theproblemofdecidingwhetheraninput word w over A representstheidentityof G (which,likethesetitrecognises, iscalledthewordproblem)canbesolvedbyaterminatingalgorithmifand onlyifthesetWP(G , A)anditscomplementcanberecognisedbyaTuring machine;thatis,ifandonlyifWP(G , A)isrecursive.Wealsopresentaproof ofthewell-knownfactthatfinitelypresentedgroupsexistforwhichtheword problemisnotsoluble;theproofofthisresultencodestheexistenceofTuring machineswithnon-recursivelanguages.
Whenthewordproblemissoluble,someconnectionscanbemadebetween thepositionofthelanguageWP(G , A)intheChomskyhierarchyandthealgebraicpropertiesofthegroup.ItiselementarytoseethatWP(G , A)isregularif andonlyif G isfinite,whileahighlynon-trivialresultofMullerandSchupp showsthatagrouphascontext-freewordproblemifandonlyifithasafree subgroupoffiniteindex.
Attachinganoutput-tapetoanautomatonextendsitfromadevicethat
Preface
definesasetofstringstoafunctionfromonesetofstringstoanother.We callsuchadeviceatransducer,andshowhowtransducerscanbeusedtodefinegroups.Amongthesegroupsarefinitelygeneratedinfinitetorsiongroups, groupsofintermediategrowth,groupsofnon-uniformexponentialgrowth,iteratedmonodromygroupsofpost-criticallyfiniteself-coveringsoftheRiemannsphere,counterexamplestothestrongAtiyahconjecture,andmanyothers.Ouraccountisbynomeanscomplete,asitconcentratesonintroducing terminology,theexpositionofsomebasictechniquesandpointerstotheliterature.
ThereisashorterbookbyIanChiswell[68]thatcoverssomeofthesame materialaswedo,includinggroupswithcontext-freewordproblemandanintroductiontothetheoryofautomaticgroups.Ouremphasisisonconnections betweengrouptheoryandformallanguagetheoryratherthancomputational complexity,butthereisasignificantoverlapbetweentheseareas.WerecommendalsothearticlebyMarkSapir[226]forasurveyofresultsconcerning thetimeandspacecomplexityofthefundamentaldecisionproblemsingroup theory.
Acknowledgements WearegratefultoProfessorRickThomasfornumeroushelpfulanddetaileddiscussionsonthetopicscoveredinthisbook.Some ofthetextinSection1.2,andthestatementandproofsofPropositions3.4.9 and3.4.10werewrittenoriginallybyRick.
WearegratefulalsotoProfessorSusanHermillerandtoananonymous reviewerforavarietyofhelpfulcomments.
Grouptheory
1.1Introductionandbasicnotation
Inthisbookweareassumingthatthereaderhasstudiedgrouptheoryatundergraduatelevel,andisfamiliarwithitsfundamentalresults,includingthe basictheoryoffreegroupsandgrouppresentations.However,inmanyofthe interactionsbetweengrouptheoryandformallanguagetheory,itisconvenient toconsidergrouppresentationsasspecialcasesofsemigroupandmonoidpresentations,sowedescribethemfromthataspecthere.
Wereferthereadertooneofthestandardtextbooksongrouptheory,such as[223]or[221]forthedefinitionsandbasicpropertiesofnilpotent,soluble (solvable)andpolycyclicgroups,
Wealsoincludesomespecifictopics,mainlyfromcombinatorialgrouptheory,thatwillberequiredlater.Thenormalformtheoremsforfreeproducts withamalgamationand HNN-extensionsareusedintheproofsoftheinsolubilityofthewordproblemingroups,andwesummarisetheirproofs.We introduceCayleygraphsandtheirmetricalproperties,andtheideaof quasiisometry betweengroups,whichplaysacentralroleintheareaandthroughout geometricgrouptheory,andwedefinethesmallcancellationpropertiesofpresentationsanddescriberelatedresults.
Thefinalsectionofthechapterisdevotedtoabriefintroductiontosomeof thespecificfamiliesofgroups,suchasCoxetergroupsandbraidgroups,that arisefrequentlyasexamplesthroughoutthebook.Theinformedreadermay prefernottoreadthischapterindetail,buttoreferbacktoitasnecessary.
1.1.1Somebasicnotation For g, h inagroup,wedefinetheconjugateof g by h,oftenwrittenas gh ,tobe hgh 1 andthecommutator[g, h]tobe ghg 1 h 1 . Butwenotethatsomeauthorsusethenotations gh and[g, h]tomean h 1 gh and g 1 h 1 gh,respectively.
Werecallthata semigroup isasetwithanassociativebinaryoperation, usuallywrittenasmultiplication,a monoid isasemigroupwithanidentity element,anda group isamonoid G inwhicheveryelementisinvertible. Weextendthemultiplicationofelementsofasemigroup S toitssubsets, defining TU = {tu : t ∈ T , u ∈ U } andwefrequentlyshorten {t }U to tU ,aswe doforcosetsofsubgroupsofgroups.
1.1.2Stringsandwords Stringsoverafinitesetareimportantforus,since theyareusedtorepresentelementsofafinitelygeneratedgroup.
Let A beafiniteset:weoftenreferto A asan alphabet.Wecalltheelements of A its letters,andwecallafinitesequence a1 a2 ··· ak ofelementsfrom A a string or word oflength k over A.Weusethesetwotermsinterchangeably.We denoteby ε thestringoflength0,andcallthisthe nullstring or emptyword Foraword w,wewrite |w| forthelengthof w Wedenoteby Ak thesetofallstringsoflength k over A,by A∗ theset(or monoid)ofallstringsover A,andby A+ theset(orsemigroup)ofallnonempty stringsover A;thatis
For w = a1 a2 ak and i ∈ N0 ,wewrite w(i)fortheprefix a1 a2 ai of w when0 < i ≤ k , w(0) = ε and w(i) = w for i > k .
Inthisbook, A oftendenotestheset X ∪ X 1 ofgeneratorsandtheirinverses foragroup G ;weabbreviate X ∪ X 1 as X ± .Inthissituation,weoftenrefer towordsin A∗ as wordsoverX eventhoughtheyarereallywordsoverthe alphabet A.
For g ∈ G ,aword w over X ofminimallengththatrepresents g iscalled a geodesicword over X ,andwedenotethesetofallsuchgeodesicwordsby G(G , X ).If w isanarbitrarywordrepresenting g ∈ G ,thenwewrite |g| or |w|G (or |g| X or |w|G , X if X needstobespecified)forthelengthofageodesicword over X thatrepresents g.Similarly,weuse v = w tomeanthatthewords v and w areidenticalasstringsofsymbols,and v =G w tomeanthat v and w representthesameelementofthegroup.
Wecallasetofstrings(i.e.asubsetof A∗ )a language;thestudyoflanguages isthetopicofChapter2.Itisconvenientatthisstagetointroducebrieflythe notationofalanguageforagroup.
1.1.3Languagesforgroups Foragroup G generatedby X ,wecallasubset of( X ± )∗ thatcontainsatleastonerepresentativeofeachelementin G a languageforG ;ifthesetcontainspreciselyonerepresentativeofeachelementwe
1.2Generators,congruencesandpresentations 5
callita normalform for G .Weshallbeinterestedinfinding good languagesfor agroup G ;clearlyweshallneedtodecidewhatconstitutesagoodlanguage. Typicallywefindgoodexamplesastheminimalrepresentativewordsundera wordorder,suchaswordlengthor shortlex, <slex ,definedbelowin1.1.4.The shortlexnormalform foragroupselectstheleastrepresentativeofeachgroup elementundertheshortlexorderingasitsnormalformword.Theset G(G , X ) ofallgeodesicwordsprovidesanaturallanguagethatisnotingeneralanormal form.
1.1.4Shortlexorderings Shortlex orderings(alsoknownas lenlex orderings) of A∗ arisefrequentlyinthisbook.Theyaredefinedasfollows.Westartwith anytotalordering < A of A.Then,for u, v ∈ A∗ ,wedefine u <slex v ifeither (i) |u| < |v| or(ii) |u| = |v| and u islessthan v inthelexicographic(dictionary) orderingofstringsinducedbythechosenordering < A of A.
Moreprecisely,if u = a1 ··· am , v = b1 ··· bn ,then u <slex v ifeither(i) m < n or(ii) m = n and,forsome k with1 ≤ k ≤ m,wehave ai = bi for i < k and ak < A bk
Notethat <slex isawell-orderingwhenever < A is,whichofcourseisthecase when A isfinite.
1.2Generators,congruencesandpresentations
1.2.1Generators If X isasubsetofasemigroup S,monoid M orgroup G , thenwedefineSgp X ,Mon X or X tobethesmallestsubsemigroup,submonoidorsubgroupof S, M or G thatcontains X .Then X iscalledasemigroup,monoidorgroup generatingset ifthatsubstructureisequalto S, M or G respectively,andtheelementsof X arecalled generators. Wesaythatasemigroup,monoidorgroupis finitelygenerated ifitpossesses afinitegeneratingset X
1.2.2Congruences If S isasemigroupand ∼ isanequivalencerelationon S, thenwesaythat ∼ isa congruence if s1 ∼ s2 , t1 ∼ t2 =⇒ s1 t1 ∼ s2 t2 .
Wethendefinethesemigroup S /∼ tobethesemigroupwithelementsthe equivalenceclasses[ s] = {t ∈ S : t ∼ s} of ∼,where[ s1 ][ s2 ] = [ s1 s2 ].
1.2.3Presentationsforsemigroups,monoidsandgroups Forasemigroup
S generatedbyaset X ,let R = {(αi ,βi ): i ∈ I } beasetofpairsofwordsfrom X + with αi =S βi foreach i.Theelementsof R arecalled relations of S .If ∼ isthesmallestcongruenceon X + containing R,and S isisomorphicto X + /∼ , thenwesaythat R isa setofdefiningrelations for S,andthatSgp X |R isa presentation for S .Inpractice,weusuallywrite αi = βi insteadof(αi ,βi ).(This isanabuseofnotationbutthecontextshouldmakeitclearthatwedonotmean identityofwordshere.)SimilarlythemonoidpresentationMon X |R defines
themonoid X ∗ / ,forwhich isthesmallestcongruenceon X ∗ containing R.
Forgroupsthesituationismarginallymorecomplicated.If G isagroup generatedbyaset X and A = X ± ,then G isisomorphicto A∗ /∼,where ∼ is somecongruenceon A∗ containing {(aa 1 ,ε), (a 1 a,ε): a ∈ X }.Wedefinea relator of G tobeaword α ∈ A∗ with α =G ε.Let R = {αi : i ∈ I } beasetof relatorsof G .If ∼ isthesmallestcongruenceon A∗ containing {(α,ε): α ∈ R}∪{(aa 1 ,ε): a ∈ X }∪{(a 1 a,ε): a ∈ X },
andif G isisomorphicto A∗ /∼,thenwesaythat R isa setofdefiningrelators for G andthat X | R isa presentation for G .Ratherthanspecifyingarelator α,sothat α representstheidentity,wecanspecifya relation β = γ (asinthe caseofmonoidsorsemigroups),whichisequivalentto βγ 1 beingarelator.
Wesaythatasemigroup,monoidorgroupis finitelypresented (or,more accurately, finitelypresentable)ifithasapresentationinwhichthesetsof generatorsanddefiningrelationsorrelatorsarebothfinite.
1.2.4Exercise Let G = X | R andlet A = X ± .Showthat G Mon A |I X ∪R, where I X = {( xx 1 ,ε): x ∈ X }∪{( x 1 x,ε): x ∈ X } and R = {(w,ε): w ∈ R}.
1.2.5Freesemigroups,monoidsandgroups If S isasemigroupwithpresentationSgp X |∅ (whichweusuallywriteasSgp X |),thenwesaythat S isthe freesemigroup on X ;weseethat S isisomorphicto X + inthiscase.Similarly,if M isamonoidwithpresentationMon X |,thenwesaythat M isthe freemonoid on X ,andweseethat M isthenisomorphicto X ∗ .If S = X + and L ⊆ S,thenSgp L = L+ ;similarly,if M = X ∗ and L ⊆ M ,thenMon L = L∗ . If F isagroupwithapresentation X |,thenwesaythat F isthe freegroup on X ;if | X | = k ,thenwesaythat F isthefreegroupof rankk (anytwofree groupsofthesamerankbeingisomorphic).Wewrite F ( X )forthefreegroup on X and F k todenoteafreegroupofrank k .
1.2.6Exercise Let G = X | R beapresentationofagroup G .Showthatthe abovedefinitionof G ,whichisessentiallyasamonoidpresentation,agrees withthemorefamiliardefinition X | R = F ( X )/R F ( X ) ,where R F ( X ) denotesthenormalclosureof R in F ( X ).
1.2.7Reducedandcyclicallyreducedwords In F ( X ),thefreegroupon X , everyelementhasauniquerepresentationoftheform w = x 1 1 x 2 2 x n n ,where n ≥ 0, xi ∈ X and i ∈{1, 1} forall i,andwherewedonothaveboth xi = xi+1 and i = i+1 forany i;inthiscase,wesaythattheword w is reduced.Each word v ∈ A∗ isequalin F ( X )toauniquereducedword w. If w isareducedwordand w isnotoftheform x 1 vx or xvx 1 forsome x ∈ X and v ∈ A∗ ,thenwesaythat w is cyclicallyreduced.Sincereplacing adefiningrelatorbyaconjugatein F ( X )doesnotchangethegroupdefined, wemay(andoftendo)assumethatalldefiningrelatorsarecyclicallyreduced words.
1.3Decisionproblems
Inhistwowell-knownpapersin1911and1912[75,76],Dehndefinedand consideredthreedecisionproblemsinfinitelygeneratedgroups,theword,conjugacyandisomorphismproblems.Whilethewordproblemingroupsisone ofthemaintopicsstudiedinthisbook,theothertwowillonlybefleetingly considered.Agoodgeneralreferenceontheseandotherdecisionproblemsin groupsisthesurveyarticlebyMiller[192].
1.3.1Thewordproblem Asemigroup S issaidtohave solublewordproblem ifthereexistsanalgorithmthat,foranygivenwords α,β ∈ X + ,decides whether α =S β.Thesolubilityofthewordproblemforamonoidorgroup generatedby X isdefinedidenticallyexceptthatweconsiderwords α,β in X ∗ or( X ± )∗ .Forgroups,theproblemisequivalenttodecidingwhetheraninput wordisequaltotheidentityelement.ThewordproblemforgroupsisdiscussedfurtherinChapter3andinPartThreeofthisbook.Examplesoffinitely presentedsemigroupsandgroupswithinsolublewordproblemaredescribed inTheorems2.9.7and10.1.1.
1.3.2Theconjugacyandisomorphismproblems Theconjugacyproblem inasemigroup S istodecide,giventwoelements x, y ∈ S,whetherthereexists z ∈ S with zx = yz.Notethatthisrelationisnotnecessarilysymmetricin x and
y,butinagroup G itisequivalenttodecidingwhether x and y areconjugate in G
Sincethewordprobleminagroupisequivalenttodecidingwhetheranelementisconjugatetotheidentity,theconjugacyproblemisatleastashardas thewordproblem,andthereareexamplesofgroupswithsolublewordproblem butinsolubleconjugacyproblem.Anumberofsuchexamplesaredescribedin thesurveyarticlebyMiller[192],includingTheorem4.8(anextensionofone finitelygeneratedfreegroupbyanother),Theorem4.11(examplesshowing thathavingsolubleconjugacyproblemisnotinheritedbysubgroupsorovergroupsofindex2),Theorem5.4(residuallyfiniteexamples),Theorem6.3(a simplegroup),Theorem7.7(asynchronouslyautomaticgroups),andTheorem 7.8(groupswithfinitecompleterewritingsystems)ofthatarticle.
Theisomorphismproblemistodecidewhethertwogivengroups,monoids orsemigroupsareisomorphic.Typicallytheinputisdefinedbypresentations, butcouldalsobegiveninotherways,forexampleasgroupsofmatrices.There arerelativelyfewclassesforwhichtheisomorphismproblemisknowntobe soluble.Theseclassesincludepolycyclicandhyperbolicgroups[232,234,72].
1.3.3Thegeneralisedwordproblem Givenasubgroup H ofagroup G ,the generalisedwordproblemistodecide,given g ∈ G ,whether g ∈ H .Sothe wordproblemisthespecialcaseinwhich H istrivial.Weshallencounter somesituationsinwhichthisproblemissolubleinChapter8.Asfortheconjugacyproblem,thesurveyarticle[192]isanexcellentsourceofexamples(in particularinTheorems5.4and7.8ofthatarticle),inthiscaseofgroupswith solublewordproblemthathavefinitelygeneratedsubgroupswithinsoluble generalisedwordproblem.
1.4SubgroupsandSchreiergenerators
Let H beasubgroupofagroup G = X ,andlet U bearighttransversalof H in G .For g ∈ G ,denotetheuniqueelementof Hg ∩ U by g.Define Z := uxux 1 : u ∈ U, x
Then Z ⊆ H . 1.4.1Theorem Withtheabovenotation,wehaveH = Z . Ourproofneedsthefollowingresult.
1.4.2Lemma LetS = {ux 1 ux 1 1 : u ∈ U, x ∈ X }.ThenZ 1 = S.
Proof Let g ∈ Z 1 ,so g = (uxux 1 ) 1 = uxx 1 u 1 .Let v := ux ∈ U .Then, sincetheelements vx 1 and u areinthesamecosetof H ,wehave vx 1 = u, and g = vx 1 vx 1 1 ∈ S .
Conversely,let g =
1 .Then vx = u,so g 1 = vxvx 1 ∈ Z and g ∈ Z 1 .
ProofofTheorem1.4.1 Let U ∩ H = {u0 }.(Weusuallychoose u0 = 1,but thisisnotessential.)Let h ∈ H .Thenwecanwrite u 1 0 hu0 = a1 ··· al forsome ai ∈ A := X ± .For1 ≤ i ≤ l,let ui := a1 ··· al .Since u 1
hu0 ∈ H ,wehave ul = u0 .Then h =G
Notethat ui+1 = a1 al+1 isinthesamecosetof H as ui ai+1 ,so ui ai+1 = ui+1 , and h =G (u
Eachbracketedtermisin Z if ai ∈ X ,andin Z 1 if ai ∈ X 1 byLemma1.4.2. So H = Z
1.4.3Corollary Asubgroupoffiniteindexinafinitelygeneratedgroupis finitelygenerated.
1.4.4Rewriting Theprocessdescribedintheaboveproofofcalculatinga word v over Z fromaword w over X thatrepresentsanelementof H iscalled Reidemeister–Schreierrewriting.Wemayclearlyomittheidentityelement fromtherewrittenword,whichresultsinawordover Y = Z \{1},which wedenoteby ρ X,Y (w).Fromtheproof,weseeimmediatelythat:
1.4.5Remark If1 ∈ U ,then |ρ X,Y (w)|≤|w|.
1.4.6Schreiergeneratorsandtransversals Theaboveset Y ofnon-identity elementsof Z iscalledthesetof Schreiergenerators of H in G .Ofcourse,this setdependson X andon U
Theset U iscalleda Schreiertransversal ifthereisasetofwordsover X representingtheelementsof U thatisclosedundertakingprefixes.Notethat suchasetmustcontaintheemptyword,andhence1 ∈ U .Bychoosingthe leastwordineachcosetundersome reductionordering of A∗ (where A = X ± ), itcanbeshownthatSchreiertransversalsalwaysexist.Reductionorderings aredefinedin4.1.5.Theyincludetheshortlexorderingsdefinedin1.1.4. ItwasprovedbySchreier[228]that,if G isafreegroupand U isaSchreier transversal,thentheSchreiergeneratorsfreelygenerate H .
Grouptheory
Thefollowingresult,knownasthe Reidemeister–SchreierTheorem,which weshallnotprovehere,providesamethodofcomputingapresentationof thesubgroup H fromapresentationofthegroup G .Notethatitimmediately impliesthecelebrated Nielsen–SchreierTheorem,thatanysubgroupofafree groupisfree.Aswithmanyoftheresultsstatedinthischapter,wereferthe readertothestandardtextbookoncombinatorialgrouptheorybyLyndonand Schupp[183]fortheproof.
1.4.7Theorem (Reidemeister–SchreierTheorem[183,PropositionII.4.1])
LetG = X | R = F/Nbeagrouppresentation,whereF = F ( X ) isthefree grouponX,andletH = E /N ≤ G.LetUbeaSchreiertransversalofEin FandletYbetheassociatedsetofSchreiergenerators.Then Y | S with S = ρ X,Y (uru 1 ): u ∈ U, r ∈ R isapresentationofH.
1.4.8Corollary Asubgroupoffiniteindexinafinitelypresentedgroupis finitelypresented.
1.5Combininggroups
Inthissectionweintroducevariousconstructionsthatcombinegroups.We leavethedetailsofmanyoftheproofsofstatedresultstothereader,whois referredto[183,ChapterIV]fordetails.
1.5.1Freeproducts Informally,the freeproductG ∗ H ofthegroups G , H is thelargestgroupthatcontains G and H assubgroupsandisgeneratedby G and H .Formally,itcanbedefinedbyitsuniversalproperty:
(i)therearehomomorphisms ιG : G → G ∗ H and ι H : H → G ∗ H ; (ii)if K isanygroupand τG : G → K , τ H : H → K arehomomorphisms, thenthereisauniquehomomorphism α : G ∗ H → K with αιG = τG and αι H = τ H .
Asisoftenthecasewithsuchdefinitions,itisstraightforwardtoprove uniqueness,inthesensethatanytwofreeproductsof G and H areisomorphic,anditisnothardtoshowthat G ∗ H isgeneratedby ιG (G )and ι H ( H ).But theexistenceofthefreeproductisnotimmediatelyclear.
Toproveexistence,let G = X | R and H = Y | S bepresentationsof G and H .Thenwecantake
G ∗ H = X ∪ Y | R ∪ S ,
where ιG and ι H arethehomomorphismsinducedbytheembeddings X → X ∪ Y and Y → X ∪ Y ;wetacitlyassumedthat X and Y aredisjoint.
Itisnotcompletelyobviousthat ιG and ι H aremonomorphisms.Thisfollows fromanotherequivalentdescriptionof G ∗ H asthesetofalternatingproducts ofarbitrarylength(includinglength0)ofnon-trivialelementsof G and H , withmultiplicationdefinedbyconcatenationandmultiplicationswithin G and H .Withthisdescription, ιG and ι H aretheobviousembeddings,and G and H arevisiblysubgroupsof G ∗ H ,knownasthe freefactors of G ∗ H .The equivalenceofthetwodescriptionsfollowsimmediatelyinamoregeneral contextfromProposition1.5.12.
Thedefinitionextendseasilytothefreeproductofanyfamilyofgroups. Thefollowingresult,whichweshallnotprovehere,isusedintheproofofthe specialcaseoftheMuller–SchuppTheorem(Theorem11.1.1)thattorsion-free groupswithcontext-freewordproblemarevirtuallyfree.
1.5.2Theorem (Grushko’sTheorem[183,IV.1.9]) ForagroupG,letd (G ) denotetheminimalnumberofgeneratorsofG.Thend (G ∗ H ) = d (G ) + d ( H ).
1.5.3Directproducts The directproductG × H oftwogroups G , H isusually definedastheset G × H withcomponent-wisemultiplication.Wegenerally identify G and H withthecomponentsubgroups,whichcommutewitheach other,andarecalledthe directfactors of G × H .Theneachelementhasaunique representationasaproductofelementsof G and H .Itcanalsobedefinedbya universalproperty:
(i)therearehomomorphisms πG : G × H → G and π H : G × H → H ; (ii)if K isanygroupand τG : K → G and τ H : K → H arehomomorphisms, thenthereisauniquehomomorphism ϕ : K → G × H with τG = πG ◦ ϕ and τ H = π H ◦ ϕ.
If G = X | R and H = Y | S arepresentations,then G × H hasthe presentation G × H = X ∪ Y | R ∪ S ∪{[ x, y]: x ∈ X, y ∈ Y }.
Wecanextendthisdefinitiontodirectproductsoffamiliesofgroupsasfollows.Let {G ω : ω ∈ Ω} beafamilyofgroups.Thenthe (full)directproduct, alsoknownsometimesasthe Cartesianproduct, ω∈Ω G ω ofthefamilyconsistsofthesetoffunctions β : Ω →∪ω∈Ω G ω forwhich β(ω) ∈ G ω forall ω ∈ Ω,wherethegroupoperationiscomponent-wisemultiplicationineach G ω ;thatis, β1 β2 (ω) = β1 (ω)β2 (ω)forall ω ∈ Ω.
Theelementsof ω∈Ω G ω consistingofthefunctions β withfinitesupport (i.e. β(ω) = 1G forallbutfinitelymany ω ∈ Ω)formanormalsubgroupof ω∈Ω G ω .Wecallthissubgroupthe restricteddirectproduct ofthefamily {G ω : ω ∈ Ω}.Itisalsosometimescalledthe directsum ofthefamilyto distinguishitfromthedirectproduct.
1.5.4Semidirectproducts Let N and H begroups,andlet φ : H → Aut( N ) bearightactionof H on N .Wedefinethe semidirectproduct of H and N , written N φ H orjust N H ,tobetheset {(n, h): n ∈ N, h ∈ H } equipped withtheproduct
Weleaveitasanexercisetothereadertoderiveapresentationof N φ H from presentationsof H and N andtheaction φ.Wenotethatsometimesthenotation H N isusedforthesameproduct.Weidentifythesubgroups {(n, 1): n ∈ N } and {(1, h): h ∈ H } with N and H ,andhence(n, h)with nh,sothatthe expressionabovereads
Thedirectproduct N × H isthespecialcasewhen φ isthetrivialaction.
Thesemidirectproductisitselfaspecialcaseofa groupextension,whichisa group G withnormalsubgroup N and G /N H .Unfortunatelyroughlyhalfof thesetofmathematiciansrefertothisasanextensionof N by H ,andtheother halfcallitanextensionof H by N .Anextensionisisomorphictoasemidirect productifandonlyif N hasacomplementin G (thatis, G hasasubgroup K , with N ∩ K = {e}, G = NK ),inwhichcaseitisalsocalleda splitextension.
Notethatwecanalsodefineasemidirectproductoftwogroups N and H , fromaleftactionof H on N .
1.5.5Wreathproducts Let G and H begroupsandsupposethatwearegiven arightaction φ : H → Sym(Ω)of H ontheset Ω.Wedefinetheassociated (full)permutationalwreathproductG H = G φ H asfollows.
Let N = ω∈Ω G ω ,wherethegroups G ω areallequaltothesamegroup G .Sotheelementsof N arefunctions γ : Ω → G .Wedefinearightaction ψ : H → Aut( N )byputting γ ψ(h) (ω) = γ (ωφ(h 1 ) )foreach γ ∈ N , h ∈ H ,and ω ∈ Ω.Wethendefine G φ H tobethesemidirectproduct N ψ H .Sothe elementshavetheform(γ, h)with γ ∈ N and h ∈ H .Asin1.5.4,weidentify {(γ, 1): γ ∈ N }, {(1, h): h ∈ H } with N and H ,andhence(γ, h)withthe product γ h.
Ifwerestrictelementsof N tothefunctions γ : Ω → G withfinitesupport,
1.5Combininggroups
thenwegetthe restrictedwreathproduct,whichweshallwriteas G R H or G Rφ H
Thespecialcaseinwhich φ istherightregularactionof H (i.e. Ω= H and hφ(h2 ) 1 = h1 h2 for h1 , h2 ∈ H )isknownasthe standard or restrictedstandard wreathproduct.Thisisthedefaultmeaningof G H or G R H whentheaction φ isnotspecified.
Finallywementionthat,ifwearegivenarightaction ρ : G → Sym(Δ), thenwecandefineanaction ψ : G φ H → Sym(Δ × Ω)bysetting(δ,ω)ψ(γ,h) = (δρ◦γ(ω) ,ωφ(h) ).Thisrightactionplaysacentralroleinthestudyofimprimitive permutationgroups,butitwillnotfeaturemuchinthisbook.
1.5.6Exercise Showthattherestrictedstandardorpermutationalwreathproduct G R H isfinitelygeneratedifboth G and H arefinitelygenerated.Verify alsothat G H isnotfinitelygeneratedunless H isfiniteand G isfinitely generated.
1.5.7Graphproducts Let Γ beasimpleundirectedgraphwithverticeslabelledfromaset I ,andlet G i (i ∈ I )begroups.Thenthe graphproduct ofthe G i withrespectto Γ canbethoughtofasthelargestgroup G generatedbythe G i suchthat[G i , G j ] = 1whenever {i, j} isintheset E (Γ)ofedgesof Γ If Xi | Ri isapresentationof G i foreach i,then ∪i∈ I Xi |∪i∈ I Ri ∪{[ xi , x j ]: xi ∈ Xi ,
isapresentationofthegraphproduct. Notethattheright-angledArtingroups(see1.10.4)canbedescribedequivalentlyasgraphproductsofcopiesof Z
1.5.8Freeproductswithamalgamation Theamalgamatedfreeproductgeneralisesthefreeproduct.Supposethat G and H aregroupswithsubgroups A ≤ G , B ≤ H ,andthatthereisanisomorphism φ : A → B. Informally,thefreeproduct G ∗ A H of G and H amalgamatedover A (via φ) isthelargestgroup P with G , H ≤ P, G , H = P,and a = φ(a)forall a ∈ A.
1.5.9Example Supposethat Γ= G , H and G ∩ H = A,where A isasubgroupofboth G and H with |G : A|≥ 3and | H : A|≥ 2.Supposealsothat Γ actsontheleftonaset Ω andthat Ω1 , Ω2 aresubsetsof Ω with Ω1 Ω2 ,such that
(1)(G \ A)(Ω1 ) ⊆ Ω2 and( H \ A)(Ω2 ) ⊆ Ω1 ; (2) A(Ωi ) ⊆ Ωi for i = 1, 2.
Then Γ G ∗ A H
Thisresultisoftenknownasthe ping-ponglemma.Itisprovedin[74, IIB.24]forthecase A = 1butessentiallythesameproofworksforgeneral A.Thereadercouldattemptitasanexercise,usingCorollary1.5.13below.
1.5.10Exercise Let Γ= SL(2, Z) = x, y with
Let G = y, H = x,and A = G
= x2 .Showthat Γ G ∗ A H by taking Ω= Z2 , Ω1 = {( x, y) ∈ Ω : xy < 0} and Ω2 = {( x, y) ∈ Ω : xy > 0}.
1.5.11Example AsaconsequenceoftheSeifert–vanKampenTheorem[189, Chapter4,Theorem2.1],weseethat,foratopologicalspace X = Y ∪ Z for which Y , Z and Y ∩ Z areopenandpath-connected,andthefundamentalgroup π1 (Y ∩ Z )embedsnaturallyinto π1 (Y )and π1 (Z ),thefundamentalgroupof X isisomorphictothefreeproductwithamalgamation π1 (Y ) ∗π1 (Y ∩Z ) π1 (Z )(see [183,IV.2]).
Formally, G ∗ A H canbedefinedbythefollowinguniversalproperty:
(i)therearehomomorphisms ιG : G → G ∗ A H and ι H : H → G ∗ A H with
ιG (a) = ι H (φ(a))forall a ∈ A; (ii)if K isanygroupand τG : G → K , τ H : H → K arehomomorphismswith
τG (a) = τ H (φ(a))forall a ∈ A,thenthereisauniquehomomorphism
α : G ∗ A H → K with αιG = τG and αι H = τ H .
Theuniquenessof G ∗ A H uptoisomorphismfollowseasily,butnotitsexistence,whichismostconvenientlyestablishedusingpresentations,asfollows.
Let G = X | R and H = Y | S bepresentationsof G and H .Foreach elementof a ∈ A,let wa and va bewordsover X and Y representing a and φ(a), respectively,andput T := {wa = va : a ∈ A}.Then,asin[183,IV.2],wedefine
G ∗ A H := X ∪ Y | R ∪ S ∪ T , anditisstraightforwardtoshow,usingstandardpropertiesofgrouppresentations,that G ∗ A H hastheaboveuniversalproperty,where ιG and ι H aredefined tomapwordsin G andin H tothesamewordsin G ∗ A H .
Notethat,inthedefinitionof T ,itwouldbesufficienttorestrict a tothe elementsofageneratingsetof A so,if G and H arefinitelypresentedand A is finitelygenerated,then G ∗ A H isfinitelypresentable.
Butwehavestillnotprovedthat G and H aresubgroupsof G ∗ A H ;that is,that ι1 and ι2 areembeddings.Wedothatbyfindinganormalformforthe
elementsof G ∗ A H .Let U and V belefttransversalsof A in G and B ∈ H , respectively,with1G ∈ U ,1 H ∈ V .Fromnowon,weshallsuppressthemaps ιG , ι H andjustwrite g ratherthan ιG (g).
1.5.12Proposition
EveryelementofG ∗ A Hhasauniqueexpressionas t1 ··· tk aforsomek ≥ 0,wherea ∈ A,ti ∈ (U \{1G }) ∪ (V \{1 H }) for 1 ≤ i ≤ k and,fori < k,ti ∈ U ⇔ ti+1 ∈ V.
Inparticular,sincedistinctelementsofGandofHgiverisetodistinct expressionsofthisform,GandHembedintoG ∗ A Hassubgroups,and G ∩ H = A = φ( A).
Proof Bydefinition,each f ∈ G ∗ A H canbewrittenasanalternatingproduct ofelementsof G and H ,andworkingfromtheleftandwritingeachsuch elementasaproductofacosetrepresentativeandanelementof A (whichhas beenidentifiedwith φ( A) = B),wecanwrite f inthespecifiednormalform.
Let Ω bethesetofallnormalformwords.Wedefinearightactionof G ∗ A H on Ω,whichcorrespondstomultiplicationontherightbyelementsof G ∗ A H . Todothis,itissufficienttospecifytheactionsof G andof H ,whichmustof courseagreeontheamalgamatedsubgroup.
Let α = t1 ··· tk a ∈ Ω and g ∈ G .If k = 0or tk ∈ V ,thenwedefine αg = t1 ··· tk tk +1 a ,where tk +1 a =G ag.Otherwise, k > 0and tk ∈ U ,andwe put αg = t1 ··· tk 1 tk +1 a ,where tk +1 a =G tk ag.Inbothcases, tk +1 ∈ U , a ∈ A andweomit tk +1 ifitisequalto1.Wedefinetheactionof H on Ω similarly. Itiseasytoseethatthesedefinitionsdoindeeddefineactionsof G and H on Ω thatagreeontheamalgamatedsubgroup A,sowecanusethemtodefine therequiredactionof G ∗ A H on Ω.Thisfollowsfromtheuniversalpropertyof G ∗ A H .Itisalsoclearfromthedefinitionthat,taking α = ε ∈ Ω,and f tobe theelementof G ∗ A H definedbythenormalformword t1 ··· tk 1 tk a,wehave α f = t1 ··· tk 1 tk a.Sotheelementsof G representedbydistinctnormalform wordshavedistinctactionson Ω,andhencetheycannotrepresentthesame elementof G ∗ A H
1.5.13Corollary Supposethatf = f1 f2 fk ∈ G ∗ A Hwithk > 0,where fi ∈ (G \ A) ∪ ( H \ B) for 1 ≤ i ≤ kand,fori < k,fi ∈ G ⇔ fi+1 ∈ H.Thenf isnotequaltotheidentityinG ∗ A H.
Conversely,supposethatthegroupFisgeneratedbysubgroups(isomorphic to)GandHwithG ∩ H = A,wherea = F φ(a) foralla ∈ A,andthatf 1 foreveryelementf = f1 f2 fk ∈ Fwithk > 0,fi ∈ (G \ A) ∪ ( H \ B) for 1 ≤ i ≤ kand,fori < k,fi ∈ G ⇔ fi+1 ∈ H.ThenF G ∗ A H.
Proof Theassumptionsensurethat,whenweput f intonormalformasdescribedintheaboveproof,theresultingexpressionhastheform t1 ··· tk a with
thesame k and,sinceweareassumingthat k > 0,thisisnottherepresentative ε oftheidentityelement.
Fortheconverse,observethatthehypothesisimpliesthatthenormalform expressionsforelementsof F describedinProposition1.5.12representdistinct elementsof F ,andsothemap α : G ∗ A H → F specifiedby(ii)ofthedefinition ofthe G ∗ A H isanisomorphism.
Aproduct f = f1 f2 fk asintheabovecorollaryiscalleda reducedform for f .Itiscalled cyclicallyreduced ifallofitscyclicpermutationsarereduced forms.Everyelementof G ∗ A H isconjugatetoanelement u = f1 fn in cyclicallyreducedform,andeverycyclicallyreducedconjugateof u canbe obtainedbycyclicallypermuting f1 fn andthenconjugatingbyanelement oftheamalgamatedsubgroup A [183,page187].
If f1 f2 ··· fk isareducedformwith k > 1and( f1 f2 ··· fk )n isnotareduced formforsome n > 0,then fk f1 cannotbereduced,andso f1 f2 ··· fk cannotbe cyclicallyreduced.Thisprovesthefollowingresult.
1.5.14Corollary [183,Proposition12.4] AnelementoffiniteorderinG ∗ A H isconjugatetoanelementofGortoanelementofH.
1.5.15 HNN-extensions Supposenowthat A and B arebothsubgroupsofthe samegroup G ,andthatthereisanisomorphism φ : A → B.Thecorresponding HNN-extension,(duetoHigman,NeumannandNeumann[141])with stable lettert , basegroupG and associatedsubgroupsA and B,isroughlythelargest group G ∗ A,t thatcontains G asasubgroup,andisgeneratedby G andanextra generator t suchthat t 1 at = φ(a)forall a ∈ A
1.5.16Example AnalogouslytoExample1.5.11,supposethatwehaveapathconnectedtopologicalspace Y withtwohomeomorphicopensubspaces U and V ,ofwhichthefundamentalgroupsembedintothatof Y ,andsupposethatwe formanewspace X byaddingahandlethatjoins U to V usingthehomeomorphismbetweenthem.ThenitisaconsequenceoftheSeifert–vanKampen Theoremthatthefundamentalgroup π1 ( X )of X isisomorphictothe HNNextension π1 (Y ) ∗π1 (U ),t ;see[183,IV.2].
Wecanalsodefinean HNN-extensionof G = X | R viaapresentation. Again,for a ∈ A,welet wa and va bewordsover X representing a and φ(a), respectively,anddefine
G ∗ A,t := X, t | R ∪ T , where T := {t 1 wa t = va : a ∈ A}.
Again,wecanrestricttheelements a in T toageneratingsetof A,so G ∗ A,t isfinitelypresentableif G isfinitelypresentedand A isfinitelygenerated.
Thereisahomomorphism ι : G → G ∗ A,t thatmapseachwordover X to thesamewordin G ∗ A,t .Onceagain,wecanuseanormalformtoprovethat ι embeds G into G ∗ A,t ,andweshallhenceforthsuppress ι andwrite g rather than ι(g).Let U and V belefttransversalsof A and B in G ,respectively,with 1G ∈ U ,1 H ∈ V
1.5.17Proposition EveryelementofG ∗ A,t hasauniqueexpressionas t j0 g1 t j1 g2 ··· gk t jk gk +1 forsomek ≥ 0,where
(i) gi ∈ Gfor 1 ≤ i ≤ k + 1 andgi 1 for 1 ≤ i ≤ k; (ii) ji ∈ Z for 0 ≤ i ≤ kandji 0 for 1 ≤ i ≤ k; (iii) for 1 ≤ i ≤ k,wehavegi ∈ Uifji > 0,andgi ∈ Vifji < 0.
Inparticular,sincedistinctelementsofGgiverisetodistinctexpressionsof thisformwithk = 0 andj0 = 0,GembedsasasubgroupofG ∗ A,t .
Proof Clearlyeach f ∈ G ∗ A,t canbewrittenas t j0 g1 t j1 g2 ··· gk t jr gk +1 for some k ≥ 0suchthat(i)and(ii)aresatisfied.If j1 > 0,thenwewrite g1 as g 1 a with g 1 ∈ U and a ∈ A and,usingtherelation t 1 at = φ(a),replace at inthe wordby t φ(a).Similarly,if j1 < 0,thenwewrite g1 as g 1 b with g 1 ∈ V and b ∈ B,andreplace bt 1 inthewordby t 1 φ 1 (b).Byworkingthroughtheword fromlefttorightmakingthesesubstitutions,wecanbring f intotherequired normalform(i.e.satisfying(i),(ii)and(iii)).
Let Ω bethesetofallnormalformwords.Wedefinearightactionof G ∗ A,t on Ω,whichcorrespondstomultiplicationontherightbyelementsof G ∗ A,t Todothis,itissufficienttospecifytheactionsof G andof t providedthat,for each a ∈ A theactionof a followedbythatof t isthesameastheactionof t followedbythatof φ(a).Infactitismoreconvenienttospecifytheactions of t and t 1 separatelyandthencheckthattheydefineinversemappings.Let α = t j0 g1 t j1 g2 gk t jk gk +1 ∈ Ω.
If g ∈ G ,thenwedefine αg := t j0 g1 t j1 g2 gk t jk
. Weneedtosubdivideintothreecasesfortheactionof t .
(a)If gk +1 A,thenwewrite gk +1 = g k +1 a with1 g k +1 ∈ U and a ∈ A,and αt := t j0 g1 t j1 g2 ··· gk t jk g k +1 t φ(a);
(b)If gk +1 ∈ A and jk 1,then αt := t j0 g1 t j1 g2 gk t jk +1 φ(gk +1 );
(c)If gk +1 ∈ A and jk = 1,then αt := t j0 g1 t j1 g2 ··· t jk 1 g k ,where g k = gk φ(gk +1 )(and gk = 1if k = 0).
Wehavethecorrespondingthreecasesfortheactionof t 1
(a)If gk +1 B,thenwewrite gk +1 = g k +1 b with1 g k +1 ∈ V and b ∈ B,and αt 1 := t j0 g1 t j1 g2 ··· gk t jk g k +1 t 1 φ 1 (b);
Another random document with no related content on Scribd:
The Project Gutenberg eBook of Fifteen years of a dancer's life
This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Fifteen years of a dancer's life With some account of her distinguished friends
Author: Loie Fuller
Author of introduction, etc.: Anatole France
Release date: November 29, 2023 [eBook #72257]
Language: English
Original publication: United Kingdom: Herbert Jenkins Limited, 1913
Credits: Tim Lindell, Debrah Thompson and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.)
*** START OF THE PROJECT GUTENBERG EBOOK FIFTEEN YEARS OF A DANCER'S LIFE ***
“W
HOSE baby is this?”
“I don’t know.”
“Well, anyway, don’t leave it here. Take it away.”
Thereupon one of the two speakers seized the little thing and brought it into the dancing-hall.
It was an odd little baggage, with long, black, curly hair, and it weighed barely six pounds.
The two gentlemen went round the room and asked each lady if the child were hers. None claimed it.
Meanwhile two women entered the room that served as dressingroom and turned directly toward the bed where, as a last resort, the baby had been put. One of them asked, just as a few minutes before the man in the dancing-hall had asked:
“Whose child is this?”
The other woman replied:
“For Heaven’s sake what is it doing there? This is Lillie’s baby. It is only six weeks old and she brought it here with her. This really is no place for a baby of that age. Look out; you will break its neck if you hold it that way. The child is only six weeks old, I tell you.”
At this moment a woman ran from the other end of the hall. She uttered a cry and grasped the child. Blushing deeply she prepared to take it away, when one of the dancers said to her:
“She has made her entrance into society Now she will have to stay here.”
From that moment until the end of the ball the baby was the chief attraction of the evening. She cooed, laughed, waved her little hands and was passed round the hall until the last of the dancers was gone.
I was that baby. Let me explain how such an adventure came about.
It occurred in January, during a very severe winter. The thermometer registered forty degrees below zero. At that time my father, my mother, and my brothers lived on a farm about sixteen miles from Chicago. When the occasion of my appearance in the world was approaching, the temperature went so low that it was impossible to heat our house properly. My mother’s health naturally made my father anxious. He went accordingly to the village of Fullersburg, the population of which was composed almost exclusively of cousins and kinsmen, and made an arrangement with the proprietor of the only public-house of the place. In the general room there was a huge cast-iron stove. This was, in the whole countryside, the only stove which seemed to give out an appreciable heat. They transformed the bar into a sleeping-room and there it was that I first saw light. On that day the frost was thick on the window panes and the water froze in dishes two yards from the famous stove.
I am positive of all these details, for I caught a cold at the very moment of my birth, which I have never got rid of. On my father’s side I had a sturdy ancestry. I therefore came into life with a certain power of resistance, and if I have not been able to recover altogether from the effects of this initial cold, I have had the strength at all events to withstand them.
A month later we had returned to the farm, and the saloon resumed its customary appearance. I have mentioned that it was the only tavern in town, and, as we occupied the main room, we had inflicted considerable hardship upon the villagers, who were deprived of their entertainment for more than four weeks.
When I was about six weeks old a lot of people stopped one evening in front of our house. They were going to give a surprise party at a house about twenty miles from ours.
They were picking up everybody en route, and they stopped at our house to include my parents. They gave them five minutes in which to get ready. My father was an intimate friend of the people whom they were going to surprise; and, furthermore, as he was one of the best musicians of the neighbourhood he could not get out of going, as without him the company would have no chance of dancing. He accordingly consented to join the party. Then they insisted that my mother go, too.
“What will she do with the baby? Who will feed her?”
There was only one thing to do in these circumstances—take baby too.
My mother declined at first, alleging that she had no time to make the necessary preparations, but the jubilant crowd would accept no refusal. They bundled me up in a coverlet and I was packed into a sleigh, which bore me to the ball.
When we arrived they supposed that, like a well-brought-up baby, I should sleep all night, and they put me on the bed in a room temporarily transformed into a dressing-room. They covered me carefully and left me to myself.
There it was that the two gentlemen quoted at the beginning of this chapter discovered the baby agitating feet and hands in every direction. Her only clothing was a yellow flannel garment and a calico petticoat, which made her look like a poor little waif. You may imagine my mother’s feelings when she saw her daughter make an appearance in such a costume.
That at all events is how I made my debut, at the age of six weeks. I made it because I could not do otherwise. In all my life everything that I have done has had that one starting-point; I have never been able to do anything else.
I have likewise continued not to bother much about my personal appearance.
MY APPEARANCE ON A REAL STAGE AT TWO YEARS AND A HALF
WHEN I was a very small girl the president of the Chicago Progressive Lyceum, where my parents and I went every Sunday, called on my mother one afternoon, and congratulated her on the appearance I had made the preceding Sunday at the Lyceum. As my mother did not understand what he meant, I raised myself from the carpet, on which I was playing with some toys, and I explained:
“I forgot to tell you, mamma, that I recited my piece at the Lyceum last Sunday.”
“Recited your piece?” repeated my mother. “What does she mean?”
“What!” said the president, “haven’t you heard that Loie recited some poetry last Sunday?”
My mother was quite overcome with surprise. I threw myself upon her and fairly smothered her with kisses, saying,
“I forgot to tell you. I recited my piece.”
“Oh, yes,” said the president, “and she was a great success, too.”
My mother asked for explanations.
The president then told her: “During an interval between the exercises, Loie climbed up on the platform, made a pretty bow as she had seen orators do, and then, kneeling down, she recited a little prayer. What this prayer was I don’t remember.”
But my mother interrupted him.
“Oh, I know. It is the prayer she says every evening when I put her to bed.”
And I had recited that in a Sunday School thronged by freethinkers!
“After that Loie arose, and saluted the audience once more. Then immense difficulties arose. She did not dare to descend the steps in the usual way. So she sat down and let herself slide from one step to another until she reached the floor of the house. During this exercise the whole hall laughed loudly at the sight of her little yellow flannel petticoat, and her copper-toed boots beating the air. But Loie got on her feet again, and, hearing the laughter, raised her right hand and said in a shrill voice: ‘Hush! Keep quiet. I am going to recite my poem.’ She would not stir until silence was restored. Loie then recited her poem as she had promised, and returned to her seat with the air of having done the most natural thing in the world.”
The following Sunday I went as usual to the Lyceum with my brothers. My mother came, too, in the course of the afternoon, and took her seat at the end of a settee among the invited guests who took no part in our exercises. She was thinking how much she had missed in not being there the preceding Sunday to witness my “success,” when she saw a woman rise and approach the platform. The woman began to read a little paper which she held in her hand. After she had finished reading my mother heard her say:
“And now we are going to have the pleasure of hearing our little friend Loie Fuller recite a poem entitled: ‘Mary had a little Lamb.’”
My mother, absolutely amazed, was unable to stir or to say a word. She merely gasped:
“How can this little girl be so foolish! She will never be able to recite that. She has only heard it once.”
In a sort of daze she saw me rise from my seat, slowly walk to the steps and climb upon the platform, helping myself up with feet and hands. Once there I turned around and took in my audience. I made a pretty courtesy, and began in a voice which resounded throughout the hall. I repeated the little poem in so serious a manner that, despite the mistakes I must have made, the spirit of it was intelligible and impressed the audience. I did not stop once. Then I
courtesied again and everybody applauded me wildly I went back to the stairs and let myself slide down to the bottom, as I had done the preceding Sunday. Only this time no one made fun of me.
When my mother rejoined me, some time after, she was still pale and trembling. She asked me why I had not informed her of what I was going to do. I replied that I could not let her know about a thing that I did not know myself.
“Where have you learned this?”
“I don’t know, mamma.”
She said then that I must have heard it read by my brother; and I remembered that it was so. From this time on I was always reciting poems, wherever I happened to be. I used to make little speeches, but in prose, for I employed the words that were natural for me, contenting myself with translating the spirit of the things that I recited without bothering much over word-by-word renderings. With my firm and very tenacious memory, I needed then only to hear a poem once to recite it, from beginning to end, without making a single mistake. I have always had a wonderful memory. I have proved it repeatedly by unexpectedly taking parts of which I did not know a word the day before the first performance.
It was thus, for instance, when I played the part of Marguerite Gauthier in La Dame aux Camélias with only four hours to learn the lines.
On the Sunday of which I have been speaking, my mother experienced the first of the nervous shocks that might have warned her, had she been able to understand, that she was destined to become the prey of a dreadful disease, which would never leave her.
From the spring which followed my first appearance at the FoliesBergère until the time of her death she accompanied me in all my travels. As I was writing this, some days before her end, I could hear her stir or speak, for she was in the next room with two nurses watching over her night and day. While I was working I would go to her from time to time, rearrange her pillows a little, lift her, give her
