Table Of ContentLondonMathematicalSocietyStudentTexts88
Groups, Languages and Automata
DEREK F. HOLT
UniversityofWarwick
SARAH REES
UniversityofNewcastleuponTyne
CLAAS E. RO¨ VER
NationalUniversityofIreland,Galway
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Contents
Preface pageix
PARTONE INTRODUCTION 1
1 Grouptheory 3
1.1 Introductionandbasicnotation 3
1.2 Generators,congruencesandpresentations 5
1.3 Decisionproblems 7
1.4 SubgroupsandSchreiergenerators 8
1.5 Combininggroups 10
1.6 Cayleygraphs 20
1.7 Quasi-isometries 23
1.8 Endsofgraphsandgroups 25
1.9 Smallcancellation 26
1.10 Someinterestingfamiliesofgroups 28
2 Formallanguagesandautomatatheory 36
2.1 Languages,automataandgrammars 36
2.2 Typesofautomata 38
2.3 Moreongrammars 50
2.4 Syntacticmonoids 51
2.5 Finitestateautomata,regularlanguagesandgrammars 52
2.6 Pushdownautomata,context-freelanguagesandgram-
mars 60
2.7 Turing machines, recursively enumerable languages
andgrammars 81
2.8 Linearly bounded automata, context-sensitive lan-
guagesandgrammars 85
v
vi Contents
2.9 Turingmachinesanddecidability 87
2.10 Automatawithmorethanoneinputword 91
3 Introductiontothewordproblem 97
3.1 Definitionofthewordproblem 97
3.2 VanKampendiagrams 98
3.3 TheDehnfunction 102
3.4 Thewordproblemasaformallanguage 105
3.5 DehnpresentationsandDehnalgorithms 110
3.6 Fillingfunctions 114
PARTTWO FINITESTATEAUTOMATAANDGROUPS 115
4 Rewritingsystems 117
4.1 Rewritingsystemsinmonoidsandgroups 117
4.2 Theuseoffsainthereductionprocess 123
5 Automaticgroups 125
5.1 Definitionofautomaticgroups 126
5.2 Propertiesofautomaticgroups 129
5.3 Shortlexandgeodesicstructures 135
5.4 Theconstructionofshortlexautomaticstructures 135
5.5 Examplesofautomaticgroups 139
5.6 Closureproperties 142
5.7 Thefalsificationbyfellowtravellerproperty 143
5.8 Stronglygeodesicallyautomaticgroups 145
5.9 Generalisationsofautomaticity 146
6 Hyperbolicgroups 150
6.1 Hyperbolicityconditions 150
6.2 Hyperbolicityforgeodesicmetricspacesandgroups 153
6.3 Thinbigons,biautomaticityanddivergence 154
6.4 HyperbolicgroupshaveDehnpresentations 158
6.5 GroupswithlinearDehnfunctionsarehyperbolic 161
6.6 Equivalentdefinitionsofhyperbolicity 164
6.7 Quasigeodesics 165
6.8 Furtherpropertiesofhyperbolicgroups 166
7 Geodesics 169
7.1 Introduction 169
7.2 Virtuallyabeliangroupsandrelativelyhyperbolicgroups 170
7.3 Coxetergroups 173
Contents vii
7.4 Garsidegroups 176
7.5 GroupswithgeodesicslyinginsomesubclassofReg 182
7.6 Conjugacygeodesics 182
8 Subgroupsandcosetsystems 184
8.1 Rationalandquasiconvexsubsetsofgroups 184
8.2 Automaticcosetsystems 189
9 Automatagroups 194
9.1 Introducingpermutationaltransducers 194
9.2 Automatagroups 199
9.3 Groupsoftreeautomorphisms 203
9.4 Dualautomata 210
9.5 Freeautomatagroups 212
9.6 Decisionproblems 213
9.7 Resolvingfamousproblems 215
PARTTHREE THEWORDPROBLEM 219
10 Solubilityofthewordproblem 221
10.1 TheNovikov–Boonetheorem 221
10.2 Relatedresults 226
11 Context-freeandone-counterwordproblems 228
11.1 Groupswithcontext-freewordproblem 228
11.2 Groupswithone-counterwordproblem 232
12 Context-sensitivewordproblems 236
12.1 Lakin’sexample 237
12.2 Somefurtherexamples 239
12.3 Fillinglength 240
12.4 Groupswithlinearfillinglength 242
13 Wordproblemsinotherlanguageclasses 249
13.1 Real-timewordproblems 249
13.2 Indexedwordproblems 254
13.3 Poly-context-freewordproblems 254
14 Theco-wordproblemandtheconjugacyproblem 256
14.1 Theco-wordproblem 256
14.2 Co-context-freegroups 257
14.3 Indexedco-wordproblems 267
14.4 Theconjugacyproblem 268
viii Contents
References 270
IndexofNotation 283
IndexofNames 284
IndexofTopicsandTerminology 287
Preface
This book explores connections between group theory and automata theory.
Weweremotivatedtowriteitbyourobservationsofagreatdiversityofsuch
connections;weseeautomatausedtoencodecomplexity,torecogniseaspects
ofunderlyinggeometry,toprovideefficientalgorithmsforpracticalcomputa-
tion,andmore.
Thebookispitchedatbeginninggraduatestudents,andatprofessionalaca-
demicmathematicianswhoarenotfamiliarwithallaspectsoftheseintercon-
nected fields. It provides background in automata theory sufficient for its ap-
plicationstogrouptheory,andthengivesup-to-dateaccountsofthesevarious
applications.Weassumethatthereaderalreadyhasabasicknowledgeofgroup
theory,asprovidedinastandardundergraduatecourse,butwedonotassume
anypreviousknowledgeofautomatatheory.
Thegroupsthatweconsiderareallfinitelygenerated.Anelementofagroup
G is represented as a product of powers of elements of the generating set X,
andhenceasastringofsymbolsfromA := X∪X−1,alsocalledwords.Many
differentstringsmayrepresentthesameelement.Thegroupmaybedefinedby
apresentation;thatis,byitsgeneratingsetXtogetherwithasetRofrelations,
fromwhichallequationsinthegroupbetweenstringscanbederived.Alterna-
tively,asforinstanceinthecaseofautomatagroups,Gmightbedefinedasa
groupoffunctionsgeneratedbytheelementsofX.
Certainsetsofstrings,alsocalledlanguages,overAarenaturallyofinterest.
WestudythewordproblemofthegroupG,namelythesetWP(G,A)ofstrings
overAthatrepresenttheidentityelement.WedefinealanguageforGtobea
languageoverAthatmapsontoG,andconsiderthelanguageofallgeodesics,
andvariouslanguagesthatmapbijectivelytoG.Wealsoconsidercombings,
defined to be group languages for which two words representing either the
sameelementorelementsthatareadjacentintheCayleygraphfellowtravel;
thatis,theyareataboundeddistantapartthroughouttheirlength.
ix
x Preface
Weconsideranautomatontobeadevicefordefininga,typicallyinfinite,set
Lofstringsoverafinitealphabet,calledthelanguageoftheautomaton.Any
stringoverthefinitealphabetmaybeinputtotheautomaton,andistheneither
acceptedifitisin L,orrejectedifitisnot.Weconsiderautomataofvarying
degreesofcomplexity,rangingfromfinitestateautomata,whichdefineregular
languages, through pushdown automata, defining context-free languages, to
Turing machines, which set the boundaries for algorithmic recognition of a
language. In other words, we consider the full Chomsky hierarchy of formal
languagesandtheassociatedmodelsofcomputation.
Finite state automata were used by Thurston in his definition of automatic
groups after he realised that both the fellow traveller property and the finite-
ness of the set of cone types that Cannon had identified in the fundamental
groups of compact hyperbolic manifolds could be expressed in terms of reg-
ularlanguages.Forautomaticgroupsaregularcombingcanbefound;useof
the finite state automaton that defines this together with other automata that
encode fellow travelling allows in particular a quadratic time solution to the
wordproblem.Word-hyperbolicgroups,asdefinedbyGromov,canbecharac-
terisedbytheirpossessionofautomaticstructuresofaparticulartype,leading
tolineartimesolutionstothewordproblem.
ForsomegroupsthesetofallgeodesicwordsoverAisaregularlanguage.
Thisistrueforword-hyperbolicgroupsandabeliangroups,withrespecttoany
generatingset,andformanyothergroups,includingCoxetergroups,virtually
free groups and Garside groups, for certain generating sets. Many of these
groupsareinfactautomatic.
ThepositionofalanguageintheChomskyhierarchycanbeusedasamea-
sureofitscomplexity.Forexample,theproblemofdecidingwhetheraninput
word w over A represents the identity ofG (which, like the set it recognises,
is called the word problem) can be solved by a terminating algorithm if and
only if the set WP(G,A) and its complement can be recognised by a Turing
machine;thatis,ifandonlyifWP(G,A)isrecursive.Wealsopresentaproof
ofthewell-knownfactthatfinitelypresentedgroupsexistforwhichtheword
problemisnotsoluble;theproofofthisresultencodestheexistenceofTuring
machineswithnon-recursivelanguages.
Whenthewordproblemissoluble,someconnectionscanbemadebetween
thepositionofthelanguageWP(G,A)intheChomskyhierarchyandthealge-
braicpropertiesofthegroup.ItiselementarytoseethatWP(G,A)isregularif
andonlyifG isfinite,whileahighlynon-trivialresultofMullerandSchupp
shows that a group has context-free word problem if and only if it has a free
subgroupoffiniteindex.
Attaching an output-tape to an automaton extends it from a device that
Preface xi
defines a set of strings to a function from one set of strings to another. We
callsuchadeviceatransducer,andshowhowtransducerscanbeusedtode-
finegroups.Amongthesegroupsarefinitelygeneratedinfinitetorsiongroups,
groupsofintermediategrowth,groupsofnon-uniformexponentialgrowth,it-
erated monodromy groups of post-critically finite self-coverings of the Rie-
mannsphere,counterexamplestothestrongAtiyahconjecture,andmanyoth-
ers. Our account is by no means complete, as it concentrates on introducing
terminology,theexpositionofsomebasictechniquesandpointerstotheliter-
ature.
ThereisashorterbookbyIanChiswell[68]thatcoverssomeofthesame
materialaswedo,includinggroupswithcontext-freewordproblemandanin-
troductiontothetheoryofautomaticgroups.Ouremphasisisonconnections
between group theory and formal language theory rather than computational
complexity,butthereisasignificantoverlapbetweentheseareas.Werecom-
mend also the article by Mark Sapir [226] for a survey of results concerning
thetimeandspacecomplexityofthefundamentaldecisionproblemsingroup
theory.
AcknowledgementsWearegratefultoProfessorRickThomasfornumer-
oushelpfulanddetaileddiscussionsonthetopicscoveredinthisbook.Some
of the text in Section 1.2, and the statement and proofs of Propositions 3.4.9
and3.4.10werewrittenoriginallybyRick.
We are grateful also to Professor Susan Hermiller and to an anonymous
reviewerforavarietyofhelpfulcomments.
1
Group theory
1.1 Introductionandbasicnotation
In this book we are assuming that the reader has studied group theory at un-
dergraduate level, and is familiar with its fundamental results, including the
basictheoryoffreegroupsandgrouppresentations.However,inmanyofthe
interactionsbetweengrouptheoryandformallanguagetheory,itisconvenient
toconsidergrouppresentationsasspecialcasesofsemigroupandmonoidpre-
sentations,sowedescribethemfromthataspecthere.
Wereferthereadertooneofthestandardtextbooksongrouptheory,such
as[223]or[221]forthedefinitionsandbasicpropertiesofnilpotent,soluble
(solvable)andpolycyclicgroups,
Wealsoincludesomespecifictopics,mainlyfromcombinatorialgroupthe-
ory, that will be required later. The normal form theorems for free products
with amalgamation and HNN-extensions are used in the proofs of the insol-
ubility of the word problem in groups, and we summarise their proofs. We
introduce Cayley graphs and their metrical properties, and the idea of quasi-
isometrybetweengroups,whichplaysacentralroleintheareaandthroughout
geometricgrouptheory,andwedefinethesmallcancellationpropertiesofpre-
sentationsanddescriberelatedresults.
Thefinalsectionofthechapterisdevotedtoabriefintroductiontosomeof
thespecificfamiliesofgroups,suchasCoxetergroupsandbraidgroups,that
arise frequently as examples throughout the book. The informed reader may
prefernottoreadthischapterindetail,buttoreferbacktoitasnecessary.
1.1.1 Somebasicnotation Forg,hinagroup,wedefinetheconjugateofg
byh,oftenwrittenasgh,tobehgh−1andthecommutator[g,h]tobeghg−1h−1.
But we note that some authors use the notations gh and [g,h] to mean h−1gh
andg−1h−1gh,respectively.
3
