some decision problems in group theory

prob-the uniform submonoid membership problem, determining the order of an elementof the group the order problem, deciding whether or not one element of the group is a non-negative power

3208081 2006

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Justin A James, Ph.D.

University of Nebraska, 2006

Advisors: Susan M Hermiller and John C Meakin

We give an algorithm deciding the generalized power problem for word hyperbolicgroups Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro, and Short showedthat elements of infinite order in word hyperbolic groups induce quasigeodesic rays inthe Cayley graph We show that a pair of quasigeodesic rays induced by two elements

of infinite order either never meet at a vertex or intersect infinitely many times, and wegive an algorithm detecting which option occurs for a given pair of quasigeodesic rays.Since solutions to the generalized power problem correspond to points of intersectionalong these rays, this decides instances of the generalized power problem involvingelements of infinite order For finite order instances, we use the algorithm deciding theword problem in word hyperbolic groups finitely many times We extend this result toobtain an algorithm deciding membership in the product of two cyclic submonoids of

a word hyperbolic group We also give an algorithm deciding membership in finitelygenerated submonoids of the free product of two finitely presented groups, providedthere is an algorithm to decide membership in the rational subsets of each factor.This extends a result of K A Mihailova, who proved that the uniform generalizedword problem is decidable in the free product of two groups if it is decidable in eachfactor Since rational membership is known to be decidable for free groups, freeabelian groups, virtually free groups, and virtually free abelian groups, our algorithmcan be used to decide membership in finitely generated submonoids of a free product

of groups with factors drawn from these classes of groups

I would like to express my gratitude to my advisors, Dr Susan Hermiller and Dr.John Meakin, for the countless hours you spent in our weekly research meetings andfor the many ways you have helped me both personally and professionally during

my time at UNL I especially want to thank Dr Hermiller for the effort she put intoreading the drafts leading up to this final document I would also like to thank theother members of my supervisory committee: Dr Mark Brittenham, Dr AllanDonsig, and Dr David Swanson I would particularly like to thank Dr Brittenhamfor his suggestions which improved the clarity of my proofs I would also like tothank Dr Brittenham and Dr Donsig for reading the penultimate draft and

providing many helpful comments I am especially grateful for the camaraderieamong the graduate students at UNL I would like to thank Josh Brown-Kramer,

Dr Charles Cusack, Dr Benton Duncan, Pari Ford, Mike Gunderson, Ned

Hummel, Eddie Loeb, Albert Luckas, Dr Matt Koetz, and Jacob Weiss for theirfriendship You have enriched my life in innumerable ways I would be remiss if Ifailed to thank my friends in the UNL Navigators and at Oak Lake Evangelical FreeChurch Without your prayers and support, I would never have made it to thispoint You truly embody Romans 12:3-13 I would also like to thank my family:Mom, Ron, Heather, Eddie, Lisa, Amity, and Eric Your love and support meanmore to me than words are able to express Lastly, I dedicate this dissertation tothe memory of my father, Tam T James, and to my God and Savior, Jesus Christ

“Blessed be the name of God forever and ever, to whom belong wisdom and might

He changes times and seasons; he removes kings and sets up kings; he gives wisdom

to the wise and knowledge to those who have understanding; he reveals deep andhidden things; he knows what is in the darkness, and the light dwells with him.”Daniel 2:20b-22 (ESV)

2.1 Conventions and Notation 7

2.1.1 Monoids and Groups 7

2.1.2 Dehn’s Problems 9

2.1.3 Generalizations of the Word Problem 10

2.1.4 Cayley Graphs and Metric Spaces 13

2.1.5 Normal Forms 16

2.2 Languages and Automata 17

2.2.1 Rational Subsets of Monoids 17

2.2.2 Rational Problems 18

2.2.3 Algorithms in Free and Free Abelian Groups 20

2.2.4 Finite State Automata 21

3 Word Hyperbolic Groups 24 3.1 δ-hyperbolic Spaces and δ-hyperbolic Groups 25

3.2 Algorithms for δ-hyperbolic Groups 29

3.2.1 The Word Problem 29

3.2.2 The Order Problem 31

3.3 The Power Problem and the Generalized Power Problem in Word Hyperbolic Groups 32

3.3.1 Quasigeodesics and Quasiconvexity 33

3.3.2 Arzel`a − Ascoli 36

3.3.3 Properties of Geodesic and Quasigeodesic Rays 40

3.3.4 The Algorithm 54

3.3.5 Extending the Algorithm to decide {u}∗· {v}∗ 59

4 Algorithms in Free Products of Groups 65 4.1 Free Products 67

4.1.1 Definition and the Word Problem 67

4.1.2 Cancellation Diagrams 71

4.2 The Uniform Submonoid Membership Problem 76

4.2.2 An Algorithm to Recognize Short Elements 814.2.3 Generating Set Completion 93

List of Figures

2.1 A directed edge in a Cayley graph 14

2.2 The Cayley graph of the free group of rank two 14

2.3 The Cayley graph of the free abelian group of rank two 15

2.4 A - a non-deterministic finite state automaton with ǫ-transitions 23

3.1 A δ-slim triangle 27

3.2 A δ-thin triangle 28

3.3 Geodesics and quasigeodesics are Hausdorff close 35

3.4 A geodesic ray associated with a quasigeodesic ray 41

3.5 Quasigeodesic and geodesic rays are Hausdorff close 42

3.6 A point on a geodesic segment close to a given point p on a quasigeo-desic ray 42

3.7 A point on the geodesic ray close to a given point p on a quasigeodesic ray 43

3.8 A geodesic connecting two points an equal distance along a pair of geodesic rays 45

3.9 The case where p = x0 46

3.10 The case where p ∈ (x0,o2] 46

3.11 The case where p ∈ (o2,y2) 46

3.12 The case where p ∈ [y2, ∞) 47

3.13 An upper bound on the distance between a pair of non-divergent qua-sigeodesic rays 48

3.14 A lower bound on the distance between points on a pair of divergent quasigeodesic rays 49

3.15 A lower bound on the distance between a point on a quasigeodesic ray and another quasigeodesic ray when the two rays are divergent 50

3.16 The triangle inequality applied to points on a pair of divergent quasi-geodesic rays 52

3.17 A pair of points along associated geodesic rays that are more than 3δ apart 53

3.18 Finding an upper bound for t2 54

3.19 A pair of vertices on non-divergent quasigeodesic rays at a distance of < K from one another 55

tance of < K from one another . 56

3.21 A sequence of pairs of vertices, each at a distance of < K from one another 56

3.22 A path from v l to u −k labeled by w−1 60

3.23 Deciding membership in the product of two cyclic submonoids in the divergent case 61

3.24 The quasigeodesic rays induced by u−1 and v in the non-divergent case 62 3.25 A word of length less than |w| labeling a path between non-divergent quasigeodesic rays 62

3.26 A translate of a path of length less than |w| between non-divergent quasigeodesic rays 63

3.27 A word of length less than |w| labeling a path from the second ray to the first ray in a non-divergent quasigeodesic pair 63

4.1 A product in M equivalent to π(P)π k (w)π(S ) 79

4.2 The transition diagram for A(V,ǫ,C2 , 1) 79

4.3 The transition diagram for A(V,Ab,ABC,2) 80

4.4 The transition diagram for A(V,ǫ,ǫ,1) 81

4.5 The tree of computations carried out by Algorithm S 84

4.6 The prefix of the product containing v1 86

4.7 The case where x1 is a syllable of ue1 87

4.8 The case where x1 is a syllable of ue2 87

4.9 The case where x1 is a syllable of ue3 88

4.10 The case where no segments lie between x l and x l+1 88

4.11 The case x l and x l+1 are in consecutive ueis and there is a cancellation between them 89

4.12 The case where x l+1 is in ugt+2 89

4.13 The path in the automaton from ǫ to the final state S1, labeled by v1 90 4.14 The cancellations forming v2 when y1 is a segment of ufm1, and v2=G y1 91 4.15 The cancellations forming v2 when y1 is a segment of ufm1 and v2 ,G y1 91 4.16 The cancellations forming v2when y1is not a segment ofufm1 and v2,G y1 91 4.17 The case where the last surviving segment is in ufN 93

4.18 The case where the last surviving segment is in ufm r for m r <N . 93

4.19 A long cancellation in a product of generators from U . 95

4.20 Replacing the long cancellation C i with a short cancellation C′ i by adding a new generator 95

4.21 A long cancellation in a product of generators from U . 97

4.22 A long cancellation in the subproduct ugj+1 u j+kg1−1 =G T1−1R−1k 1 98

Chapter 1

Introduction

In 1911, Dehn formulated three fundamental decision problems for groups: the wordproblem, the conjugacy problem, and the isomorphism problem [Deh] Dehn proved

that there are algorithms to decide each of these problems when G is the fundamental

group of a closed 2-manifold However, these problems are undecidable for finitely

generated groups in general The word problem for a group G with finite generating set A asks whether or not there is an algorithm that, given input a word w ∈ (A∪A−1)∗,

determines in a finite number of steps whether or not w represents the trivial element

in the group In 1955, Novikov [Nov] and Boone [Boo] independently proved thatthere is a finitely generated group with undecidable word problem Since then, therehas been much progress toward the goal of determining which classes of groups havedecidable word problem and which have undecidable word problem, but many openquestions remain

If we restrict our attention to finitely generated groups with decidable word lem, there are several generalizations of the word problem that can be considered.These include: deciding membership in finitely generated subgroups (the uniformgeneralized word problem), deciding membership in finitely generated submonoids

prob-(the uniform submonoid membership problem), determining the order of an element

of the group (the order problem), deciding whether or not one element of the group

is a non-negative power of another element (the power problem), deciding, given two

elements of the group via two words u and v, whether or not there are nonzero integer exponents k and l such that u k =G v l (the generalized power problem), and decid-ing membership in rational subsets of the group (the rational membership problem).Each of these problems is known to be strictly harder than the word problem That

is, an algorithm deciding any of these problems for a finitely generated group G

im-plies that there is also an algorithm to decide the word problem, and for each of thesedecision problems, there is an example of a finitely generated group for which theword problem is decidable, but the given problem is undecidable (references for theseexamples will be given in Chapter 2) Consequently, each of these problems is unde-cidable in general However, there are examples of classes of groups for which theseproblems are decidable For example, each of these problems is decidable for both fi-nitely generated free groups and finitely generated free abelian groups In 1999, ZephGrunschlag [Gru] showed that the word problem, uniform generalized word problem,and rational membership problem are all preserved when passing to finite index sub-groups and to finite index extensions Therefore, each of these membership problems

is decidable for virtually finitely generated free groups and virtually finitely ated free abelian groups as well This dissertation explores the decidability of theaforementioned generalizations of the word problem The main results are as follows:

gener-Corollary 3.3.28 Let G = hA | Ri be a finitely generated δ-hyperbolic group with

A closed under inversion Let u, v ∈ A∗ be words representing elements of infinite

order in G Let β u and βv be the quasigeodesic rays based at 1 induced by u and v

respectively Then there is an algorithm that decides whether the pair of rays βu and

βv are divergent or non-divergent

Theorem 3.0.13 Let G = hAi be a finitely generated δ-hyperbolic group with A

closed under inversion Then there is an algorithm deciding the generalized power

problem for G.

Theorem 3.0.14 Let G = hAi be a finitely generated δ-hyperbolic group with A closed under inversion Let u, v ∈ A∗ Then there is an algorithm deciding member-

ship in the rational subset {u}∗· {v}∗

Theorem 4.2.9 Let G = G1∗ G2 where G1 and G2 are finitely presented groups with

generating sets A and B respectively, both closed under inversion If G1 and G2 havedecidable rational membership problem, then the uniform submonoid membership

problem for G is decidable.

Here is an outline of the material in this dissertation:

Chapter 2 introduces the notation and background concepts necessary for the gorithms developed in the remainder of the dissertation It outlines the historicaldevelopment of the word problem and its generalizations, and discusses examples offinitely generated groups for which these decision problems are decidable and othersfor which they are undecidable Some tools of particular importance that are de-veloped in this chapter are Cayley graphs, rational subsets, normal forms and finitestate automata

al-In Chapter 3, we turn our attention to algorithms in word hyperbolic groups.Word hyperbolic groups were introduced by Gromov in 1987 [Gro] It is known thatthe word problem [A+], the order problem ([Lys], [Bra]), and the power problem[Lys] are decidable for word hyperbolic groups However, E Rips [Rip] showed thatthe uniform generalized word problem is undecidable in general for word hyperbolicgroups Therefore, membership in finitely generated submonoids and in rational sub-sets are also undecidable We prove that the generalized power problem is decidablefor word hyperbolic groups To do so, we prove some geometric and algorithmic prop-

erties of the quasigeodesic rays induced by elements of infinite order in the Cayleygraph of a word hyperbolic group Our main result is that pairs of quasigeodesic rays

in the Cayley graph of a word hyperbolic group induced by elements of infinite ordereither never meet at a vertex, or they intersect at a vertex infinitely many times, and

we can detect which of the two happens for a given pair of quasigeodesic rays Sincesolutions to the generalized power problem for pairs of words representing elements ofinfinite order correspond to points of intersection on their induced quasigeodesic rays,this result leads to an algorithm to decide the generalized power problem in instanceswhere both words represent elements of infinite order The case where the givenwords represent elements of finite order can be decided using the algorithm decidingthe word problem for word hyperbolic groups finitely many times The algorithmdeciding the order problem for word hyperbolic groups can be used to determinewhether we should use the infinite order algorithm or the finite order algorithm for agiven pair of test words Together, these give an algorithm deciding the generalizedpower problem for word hyperbolic groups We use similar techniques to extend this

to an algorithm deciding membership in the product of two cyclic submonoids of aword hyperbolic group

In Chapter 4, we shift our focus to another perspective on decision problems ingroups Rather than considering a class of groups, we consider the effect of classicalgroup theoretic constructions on decision algorithms The most common construc-tions for building new groups from old are: subgroups, quotients, direct products,free products, free products with amalgamation, and HNN extensions It is natural

to ask what effect these constructions have on the decidability of the word problemand its generalizations A theorem of Grunschlag [Gru] shows that the word prob-lem, uniform generalized word problem, and rational membership problem are allpreserved when passing to finite index subgroups and finite index extensions K A

Mihailova has shown that the uniform generalized word problem is not preserved bydirect products [Mih1], but that it is preserved by free products [Mih3].

Most of chapter 4 is devoted to extending Mihailova’s result on the uniform eralized word problem for free products to obtain a similar result for the uniformsubmonoid membership problem Using the normal form theorem for free products,

gen-we define the cancellation diagram of a product of generators in a finitely generatedsubmonoid of a free product of two groups with decidable word problem (these aresimilar to the cancellation diagrams introduced by Patrick Bahls [Bah]) These di-agrams allow us to keep track of exactly how cancellation occurs when putting aproduct of generators in a finitely generated submonoid of the free product of twogroups into reduced alternating form The cancellation diagram for a product in thesubmonoid generators is short if each of its maximal cancellations is short, that is, ifeach maximal cancellation involves at most three consecutive submonoid generators

in the product (see Definition 4.1.12) We give an automaton construction that formsthe foundation of an algorithm that recognizes the elements of a finitely generatedsubmonoid that can be written as short products We then give a procedure that,

given a finite generating set U for a submonoid M of a free product G = G1 ∗ G2,

forms a finite generating set U with the property that every element of the monoid can be written as a short product with respect to U The completion procedure is

conceptually similar to the procedure for forming a Nielsen closed generating set for

a submonoid of a free group Once we have the generating set U, the algorithm that

recognizes elements of the monoid that can be written as a short product with respect

to a given finite generating set can be used to decide membership in the submonoid,since every word that represents an element in the monoid can be written as a short

product in U The idea of deciding membership by first finding an algorithm that

recognizes short elements and then finding a generating for which every element is

short is due to Mihailova [Mih4] However, since we are deciding membership in monoids rather than subgroups, our algorithm recognizing short elements requiresdifferent techniques than those employed by Mihailova.

sub-Notice that Mihailova’s result for the uniform generalized word problem in a freeproduct only requires an algorithm to decide the uniform generalized word problem

in each factor, while our algorithm deciding uniform submonoid membership in afree product requires an algorithm to decide membership in rational subsets in eachfactor, which is a stronger hypothesis than just requiring an algorithm to decideuniform submonoid membership in each factor

Chapter 2

Preliminaries

The purpose of this chapter is to introduce some standard notation that will beused throughout this dissertation and the background material necessary to make theresults in the chapters that follow accessible

2.1 Conventions and Notation

Let A be a set The free monoid on A, denoted by A∗, is the set of all finite length

strings (or words) over A with concatenation as its product and the empty string

ǫ as its unit A language L is any subset of A∗

Given a word w = a1a2 an with

a i ∈ A for all i, the length of w, denoted by |w|, is n, the number of letters in w By convention, l(ǫ) = 0.

A set A is a generating set for a monoid M if there is a surjective homomorphism

π : A∗ → M Similarly, A is a generating set for a group G if there is a surjective homomorphism π : (A ∪ A−1)∗ → G, where A−1

:= {a−1 | a ∈ A} is a set of formal

inverses corresponding to A and π(a−1

) = π(a)−1 for every a ∈ A A generating set A

is closed under inversion if A = B ∪ B−1 where B is a generating set for G, and

as u ≡ v Given U ⊆ A∗, hUi denotes the subgroup of G generated by U (formally speaking, we mean the subgroup of G generated by π(U)) Similarly, hUi N denotes

the normal subgroup of G generated by U, and MonhUi denotes the submonoid of G generated by U

Given any non-empty set X, we can construct F(X), the free group on the set X Let X−1

:= {x−1 | x ∈ X} We define an equivalence relation on (X ∪ X−1)∗ by taking

two words w and v to be equivalent, written w ∼ v, if it is possible to pass from one word to the other by inserting and deleting subwords of the form xx−1 or x−1x for

x ∈ X A word w ∈ (X ∪ X−1)∗ is reduced if it contains no subwords of the form xx−1

or x−1x Every equivalence class [w] contains exactly one reduced word (see [Rob] p 46) Then F(X) := {[w] | w ∈ (X ∪ X−1)∗}, is a group with operation [w][v] := [wv] (the equivalence class of the word formed by concatenating w and v), identity element [ǫ] (the equivalence class if the empty word), and inverses [w]−1

is called a set of relators if r = G 1 for every r ∈ R.

P = hA | Ri is called a presentation for a group G if A is a generating set for G, and the natural homomorphism ϕ : F(A)/hRi N → G is an isomorphism

A group G is finitely presented if there is a presentation P = hA | Ri with

|A| < ∞, |R| < ∞, and G  F(A)/hRi N

Example: The free abelian group of rank n is the group with presentation Z n :=

ha1,a2, ,a n | a i a j a−1i a−1j = 1 for 1 ≤ i < j ≤ ni.

Since w1 =G w2 if and only if w1w−1

2 =G 1, this is equivalent to deciding for a given

word w ∈ (A ∪ A−1)∗, whether or not w = G1 If such an algorithm exists, then we say

the group G with generating set A has decidable word problem Otherwise, we say (G, A) has undecidable word problem.

(2) Given a group G with generating set A, the conjugacy problem asks if there

is an algorithm that decides, given two words w1,w2 ∈ (A ∪ A−1)∗, whether or not w1

and w2 represent conjugate elements in G; that is, whether or not there is some g ∈ G such that g−1π(w1)g = π(w2) If such an algorithm exists, then we say the pair (G, A) has decidable conjugacy problem Otherwise, we say (G, A) has undecidable

Proposition 2.1.2 ([LS] Proposition 2.2 p 90) If G = hA | Ri is a presentation

with finite generating set for which the word problem or the conjugacy problem isdecidable, then the word problem (or the conjugacy problem) is decidable for anyfinitely generated presentation of the group

(1) In light of the previous proposition, it makes sense to say that a group G

has decidable word problem (or conjugacy problem) without reference to a specificgenerating set, with the understanding that this means that the word (or conjugacy)problem is decidable for any finite generating set for the group

(2) Dehn showed that all three of his decision problems are decidable when G is the

fundamental group of a closed 2-manifold ([Deh]) However, the word problem turnsout to be undecidable in general That is, there are examples of finitely generatedgroups with undecidable word problem (see [Nov], [Boo], and [Hig])

(3) If G has decidable conjugacy problem, it also has decidable word problem To see this, let G be a group with finite generating set A, with A closed under inversion Let w ∈ A∗

Notice that w = G 1if and only if g−1π(w)g = G1 for some g ∈ G Then we

can use the algorithm deciding the conjugacy problem for this presentation in order

to check to see if w and 1 are conjugate in G, hence we can decide the word problem for G.

(4) There is a finitely generated group G with decidable word problem and

unde-cidable conjugacy problem ([Fri], [Col])

(5) The isomorphism problem is undecidable In fact, there is no algorithm that,given a finite presentation, decides whether or not the group given by this presentation

is the trivial group ([Adi1], [Adi2], [Rab])

In this section, we consider several generalizations of the word problem Each problem

is known to be a strictly harder problem than the word problem for the class of finitelygenerated groups

Definition 2.1.3 Let G = hAi be a finitely generated group, with A closed under

inversion The uniform generalized word problem asks if there is an algorithm

that decides, given a finite collection of words u1,u2, ,u k ∈ A∗and a test word w ∈ A∗,

whether or not π(w) ∈ H := hπ(u1), π(u2), , π(u k)i If such an algorithm exists, we say

G has decidable uniform generalized word problem Otherwise, we say G has

undecidable uniform generalized word problem

Definition 2.1.4 Let G = hAi be a finitely generated group, with A closed under

inversion The uniform submonoid membership problem asks if there is an

algorithm that decides, given a finite collection of words u1,u2, ,u k ∈ (A ∪ A−1)∗ and

a test word w ∈ (A ∪ A−1)∗, whether or not π(w) ∈ M := Monhπ(u1), π(u2), , π(u k)i

If such an algorithm exists, we say G has decidable uniform submonoid

mem-bership problem Otherwise, we say G has undecidable uniform submonoidmembership problem

Definition 2.1.5 Let G = hAi be a finitely generated group, with A closed under

inversion The order problem asks if there is an algorithm that, given a word

w ∈ A∗, determines whether or not π(w) has finite order in G If π(w) has finite order, the algorithm outputs the order of π(w) (that is, the least n ∈ N+ such that w n

=G1)

On the other hand, if π(w) has infinite order, the algorithm stops after a finite number

of steps and outputs ∞ If such an algorithm exists, we say G has decidable order

problem Otherwise, we say G has undecidable order problem.

Definition 2.1.6 Let G = hAi be a finitely generated group, with A closed under

inversion The power problem asks if there is an algorithm that, given two words

u, v ∈ A∗, determines whether or not there is an integer k ∈ N such that u = G v k If

such an algorithm exists, we say G has decidable power problem Otherwise, we say G has undecidable power problem.

Definition 2.1.7 Let G = hAi be a finitely generated group, with A closed under

inversion The root problem asks if there is an algorithm that, given any word

w ∈ A∗and any integer k, determines whether or not w has a kth root That is, whether

or not there is an element g ∈ G such that π(w) = G g k If such an algorithm exists, we

say G has decidable root problem Otherwise, we say G has undecidable root

problem

Definition 2.1.8 Let G = hAi be a finitely generated group, with A closed under

inversion The generalized power problem asks if there is an algorithm that,

given two words u, v ∈ A∗, determines whether or not there are nonzero integers

k, l ∈ Z such that u k

=G v l If such an algorithm exists, we say G has decidable

generalized power problem Otherwise, we say G has undecidable generalizedpower problem

Proposition 2.1.9 ([LM] Theorem 1 p 8): If the uniform generalized word problem,the uniform submonoid membership problem, the order problem, the root problem, thepower problem, or the generalized power problem are decidable for a finitely generated

group G, then the word problem for G is also decidable.

Notes:

(1) There is a finitely generated group for which the word problem is decidable,

and the uniform generalized word problem is undecidable In fact, we can take G to

be F × F where F is a free group of rank at least 2 ([Mih1])

(2) Since every finitely generated subgroup of a group G is also a finitely generated

submonoid, any algorithm deciding the uniform submonoid membership problem for

G can also be used to decide the uniform generalized word problem for G Hence the

uniform submonoid membership problem is at least as hard as the uniform generalizedword problem

(3) The order and power problems were introduced by J McCool ([Mc1]) McCool

gives an example of a finitely generated group G with decidable word problem and

undecidable order and power problems (Theorem 2 in [Mc2] p 837)

(4) The root problem and the generalized power problem were introduced by S.Lipschutz and C F Miller ([LM]) Their paper gives many examples that demon-strate the relationships between generalizations of the word problem including: anexample of a finitely generated group with decidable generalized power problem andundecidable order, power, and root problems, and an example of a finitely generatedgroup with decidable order problem and undecidable power, root, and generalizedpower problems (see [LM] pp 8-9)

(5) If the uniform submonoid membership problem is decidable for a finitely

gen-erated group G, then the power problem is also decidable for G, since the power

problem is equivalent to deciding membership in a cyclic submonoid

Question 2.1.10 Is there a finitely generated group G with decidable uniform

gen-eralized word problem and undecidable uniform submonoid membership problem?

Definition 2.1.11 Let G be a group and A a generating set for G that is closed under inversion The Cayley graph of G with respect to A, denoted by Γ (G,A), is the directedlabeled graph defined as follows:

(1) V(Γ (G,A) ) = G That is, there is a vertex for each element of G.

(2) For every vertex g and every a ∈ A, there is a directed edge (g, ga) from g to

ga labeled by a (see Figure 2.1).

Note: Since A is closed under inversion, we can write A = B ∪ B−1 For each directed

edge (g, gb) labeled by b ∈ B, there is an inverse edge (gb, g) labeled by b−1 ∈ B−1

a a

b

b

a b b a

b

a a

a

b

Figure 2.2: The Cayley graph of the free group of rank two

Therefore, we can omit the inverse edges labeled over B−1, yielding a directed graph

labeled by B, with the understanding that the B labeled edge is represented by moving along the directed edge, and the B−1 labeled inverse edge is represented by movingopposite the arrow along the directed edge

b a

b b b

b

a a

b b

Figure 2.3: The Cayley graph of the free abelian group of rank two

single undirected edge Each of these undirected edges is taken to be isometric to theunit interval [0, 1] in R The distance between arbitrary points in the graph (includingpoints in the interior of edges) is defined to be the length of the shortest path in thegraph connecting them Since there is a one to one correspondence between elements

of the group and vertices in the Cayley graph, the metric on the Cayley graph induces

a metric on the group G For any pair of elements g, h ∈ G, d(g, h) is the distance

between the vertices in Γ(G,A) corresponding to the elements g and h.

(2) Since Γ(G,A) is a labeled graph, when viewed as a directed graph, each path of

minimal length between two vertices g and h is labeled by a word in A∗ Any word in

A∗ that is the label of a minimal length path is called a geodesic word For a fixed

element g ∈ G, the collection of all words labeling minimal length paths from vertex

1to vertex g is the set of geodesic representatives of g with respect to the generating set A.

2.1.5 Normal Forms

Definition 2.1.12 Let G be a group with finite generating set A, and suppose A is closed under inversion Let π : A∗→ G be the canonical surjective homomorphism Anormal form for G is a subset L ⊆ A∗ such that π| L : L → G is a bijection

Note: We will usually restrict our attention to the case where, given a word w ∈ A∗

there is an algorithm to find the normal form representing the corresponding group

element That is, given w, one can find bw ∈ L such that π(w) = π(b w).

Suppose A = {a1,a2, ,a n}, and let w ∈ A∗ Let w ≡ s1s2 sm with each s i ∈ A Recall that the length w, written |w| := m, is the number of letters in w Any fixed ordering on A induces a shortlex ordering on A∗ as follows: given u, v ∈ A∗, suppose

that u ≡ x1x2 x m and v ≡ y1y2 yn We say u is shortlex less than v (written u < sl v) if: (1) m < n or (2) m = n, x i = y i for i < k, and x k <y k in the order on A.

Since we will never use two different orderings of A at the same time, we can omit reference to the ordering on A in our notation < sl without fear of ambiguity

Definition 2.1.13 A shortlex normal form for G is a normal form L for which there is an order on A such that for any w ∈ L and v ∈ A∗

representative of w with respect to a fixed ordering on A, then the word problem for

G is decidable To see this, notice that w = G 1 if and only if bw = ǫ The following

proposition shows that the converse to this statement is also true:

Proposition 2.1.14 Let G = hA | Ri be a finitely generated group with A closed under inversion If the word problem for G with respect to this generating set is decidable,

then there is an algorithm that, given a word w ∈ A∗, outputs w, the shortlex normalb

form for w with respect to a fixed ordering on the generating set A.

Proof: Given w ≡ x1x2 xm and some fixed ordering on A, let S := {v ∈ A∗ | v < sl w}, the set of words shorter than w in the related shortlex ordering Notice that S is a finite subset of A∗, hence S is recursive We can effectively put S in decreasing order

with respect to <sl For each v ∈ S , apply the algorithm deciding the word problem

to test to see if wv−1

=G 1 If the answer is NO for every v ∈ S , then w is in shortlex normal form Otherwise, let v N be the least element of S such that the word problem algorithm returns YES Then v N is the shortlex normal form for w. 

2.2 Languages and Automata

Let M be a monoid Then Rat(M), the rational subsets of a monoid M, is the smallest family of subsets of M containing the empty set and every singleton set, and

closed under the operations: finite union, finite product, and submonoid closure

A convenient way to represent rational subsets is by means of regular sions Regular expressions are recursively defined as follows:

expres-(i) ∅ is a regular expression and denotes the empty set

(ii) for each m ∈ M, m is a regular expression and denotes the set {m}.

(iii) If r and s are regular expressions denoting rational subsets R and S tively, then (r ∪ s), (rs), and (r∗) are regular expressions that denote R ∪ S , RS , and

respec-R∗ respectively

Given a regular expression r, R(r) denotes the rational subset of M associated with r.

Note: Distinct regular expressions are sometimes associated with the same rational

subset of M.

Proposition 2.2.1 ([Ber] Prop 2.2 p 56) Let M, M′ be monoids, and let α : M → M′

be a homomorphism If U ∈ Rat(M), then α(U) ∈ Rat(M′) Furthermore, if α is

surjective, then for any U′ ∈ Rat(M′) there is a set U ∈ Rat(M) such that α(U) = U′

Definition 2.2.2 Let M be a monoid with generating set A The rational

member-ship problem asks if there is an algorithm that decides, given a rational expression

r and a test word w ∈ A∗, whether or not π(w) ∈ R(r) If such an algorithm exists, we say M has decidable rational membership problem Otherwise, we say M has

undecidable rational membership problem

Definition 2.2.3 A rational problem is an algorithmic problem that can be decidedusing a finite number of instances of the rational membership problem

Notes:

(1) The word problem is a rational problem, since it can be decided by determiningwhether or not π(ǫ) is in the rational subset associated with the rational expression

π(w)

(2) The order problem is a rational problem since given a test word w, we can

find its order by first deciding whether or not ǫ is in the rational subset associated

with π(w)(π(w))∗ If not, w has infinite order Otherwise, it has finite order, and we

can determine its order by successively checking to see if π(ǫ) is in the rational subset

associated with π(w k) for each k ∈ N The first k for which the answer is yes is the order of π(w).

(3) Let G be a finitely generated group with finite generating set A, with A closed under inversion, and let U = {u1,u2, ,u k} be a finite collection of words with each

u i ∈ A∗ Then both the uniform submonoid membership problem and the uniform

generalized word problem are rational problems since they can be decided by

deter-mining whether or not a test word w ∈ A∗ is in the rational subset associated with

(4) The generalized power problem is a rational problem, since given pair of words

u, v, there are non-zero integers k and l such that u k =G v l if and only of the intersection

of the rational subsets associated with π(u)(π(u))∗and π(v)(π(v))∗ is non-empty, which

is a rational problem (see [Gru] p 43)

(5) Although the rational membership problem is the most general of the decisionproblems that we have considered so far, there are classes of groups for which thisproblem is known to be decidable

Theorem 2.2.4 ([Ben]) Let F be the free group on a finite set X Then the rational membership problem is decidable for F.

Theorem 2.2.5 ([Gru] p 53) The rational membership problem is decidable for Zn

We close our discussion on rational problems with a theorem of Grunschlag thatshows that the ability to decide membership in rational subsets of a finitely generated

group G is inherited by finite index subgroups of G and can also be lifted to finite index extensions of G.

Definition 2.2.6 Let H be a subgroup of a group G The index of H in G, denoted

[G : H] , is the number of left cosets of H in G If [G : H] < ∞, then we say H has finite index in G Two groups G1 and G2 are commensurable if there are

subgroups H1<G1 and H2<G2 such that [G1 : H1] < ∞, [G2 : H2] < ∞, and H1  H2

Theorem 2.2.7 ([Gru] Theorem 2.3.3, p 57) Let G and H be finitely generated infinite groups that are commensurable Then the rational membership problem for G

is decidable if and only if it is decidable for H.

2.2.3 Algorithms in Free and Free Abelian Groups

As mentioned above, the fact that membership in rational subsets is decidable forfinitely generated free groups and finitely generated free abelian groups implies thatone can also decide the word problem and each of the generalizations of the wordproblem mentioned above There are also previously known algorithms that can

be used to decide many of the generalizations of the word problem in these classes

of groups Since the techniques employed in these algorithms help to motivate thealgorithm in Chapter 4, we will discuss them briefly

The word problem for any finitely generated free group F(X) can be decided by putting any input word w into reduced form A word w = G 1if and only if its reducedform is the empty word (see [MKS] p 397) Since the set of all reduced words is a

shortlex normal form for F(X), we will use wbto denote the unique reduced word in the

equivalence class [w] The uniform generalized word problem in any finitely generated

free group can be decided using a finite state automaton construction (see [Sta] or[BMMW]) We will define finite state automata in the section that follows Theuniform generalized word problem for a finitely generated free group was originallysolved using an algorithm that finds a Nielsen closed generating set for any finitelygenerated subgroup (see [Nie], or [LS] pp 4-7) An algorithm due to Ivanov, Margolis,and Meakin extends Nielsen’s methods to decide the uniform submonoid membershipproblem for free groups We will discuss their algorithm briefly (see [IMM] Lemma5.4, pp 105-106 for more details)

Definition 2.2.8 A generating set V for a finitely generated submonoid M of a free group F is Nielsen closed if it satisfies the following two properties:

(N0) Every element of V is a reduced word.

(N1) If v1,v2 ∈ V and v1v2 ,F 1, then either |vd1v2| > max{|v1|, |v2|} or vd1v2 ∈ V.

With this in mind, we define the following operations on a generating set V for a submonoid of a free group F:

(T1) For i , j, if v i,v j ∈ V, 0 < |dv i v j | ≤ max{|v i |, |v j|} and dv i v j <V, add vdi v j to V (T2) Delete v i if v i ∈ V and bv i ≡vbj for i , j.

We can obtain a Nielsen closed generating set U for any finite generating set V for a submonoid M by first replacing elements of the initial generating set V with

their normal forms, and then iteratively applying the operations (T1) and (T2) If

[w] ∈ M, with |bw| = k, then [w] can be written as a product [w] = G

and each uej ∈ U Therefore, membership in the submonoid M can be decided using

finitely many instances of the word problem, which is decidable

Note: In the algorithm deciding the uniform submonoid membership problem bership for free products of groups given in Chapter 4, we use a procedure to “com-plete” a finite generating set in order to obtain a “closed” generating set for whichevery element of the monoid can be expressed as a “short” product

mem-We now turn our attention to algorithms in free abelian groups The word problem

is decidable in Zn for any n since there is an algorithm to reduce any word w ∈

(A ∪ A−1)∗, where A = {a1,a2, ,a n}, to a word v of the form a z1

1a z2

2 az n

n , z i ∈ Z(c.f [E+]

Theorem 4.3.1 p 96) satisfying w =Zn v Membership in finitely generated submonoids

can be decided by reducing the problem to non-negative integer programming, which

is known to be decidable (In fact, this problem is NP-Complete (see [GJ] p 245))

Except where noted otherwise, the material in this section is from [HU] Chapter 2

Definition 2.2.9 A nondeterministic finite state automaton (NFA) with

ǫ-transitions A is denoted by a 5-tuple (Q, A, δ, q0,F) Here, Q is a finite set of states,

A is a finite input alphabet, q0 ∈ Q is the initial state, F ⊆ Q is the set of final states, and δ is a transition function mapping Q × (A ∪ {ǫ}) to 2 Q , the set of all subsets of Q.

We can extend δ to a function ˆδ from Q × A∗ to Q as follows: For each state q ∈ Q, let ǫ − CLOS URE(q) denote the set of all states p such that there is a sequence

of transitions from q to p labeled by ǫ For any P ⊆ Q, let ǫ − CLOS URE(P) :=

S

q∈Pǫ −CLOS URE(q) Then we define:

(i) ˆδ(q, ǫ) = (ǫ − CLOS URE(q)).

(ii) ˆδ(q, wa) = (ǫ −CLOS URE(P)), where P = {p | for some r in ˆδ(q, w), p ∈ δ(r, a)} (iii) For R ⊆ Q, ˆδ(R, a) =S

dia-L(A), the language accepted by A, as {w ∈ (A ∪ {ǫ})∗ | ˆδ(q0,w) contains a state in F}.

A word w ∈ L(A) if and only if there is a directed path in the transition diagram of

Afrom the initial state to some final state labeled by w Given a word w ∈ (A ∪ {ǫ})∗,

by mapping each ǫ in w to the empty word, we obtain a word w′ ∈ A∗ Let L′(A),

the language over A∗ accepted by A, be {w′ ∈ A∗ | w ∈ L(A)}.

Definition 2.2.10 ([Ber] p 16) A language L′ ⊆ A∗ is recognizable if it is L′(A)

for some FS A A The family of all such languages is denoted Rec(A∗)

Definition 2.2.11 ([Ber] p 52) More generally, let M be a monoid A subset L′ of

M is recognizable if there is a finite monoid N, and homomorphism α : M → N,

Figure 2.4: A - a non-deterministic finite state automaton with ǫ-transitions.

and a subset T ⊆ N such that L′

= α−1(T ) The collection of all recognizable subsets

of M is denoted by Rec(M).

The following theorem shows that for free monoids, the classes of languages describedover the past two sections all coincide:

Theorem 2.2.12 (Kleene) (see [HU] pp 29-34 and [Gru] p 18) Let A∗ be the free

monoid over a finite alphabet A Let L′ ⊆ A∗ Then the following are equivalent:(i) L′ ∈ Rec(A∗)

(ii) L′ is R(r) for some regular expression r.

(iii) L′ is the language of some NFA with ǫ-transitions.

(iv) L′ ∈ Rat(A∗)

(v) There is a finite monoid N, a homomorphism α : A∗→ N, and a subset T ⊆ N

such that L′

= α−1(T )

Note: For an arbitrary finitely generated monoid M, Rec(M) ⊆ Rat(M), but equality

does not hold in general (for more details, see [Ber] p 51)

Chapter 3

Word Hyperbolic Groups

Let G = hAi for some finite set A that is closed under inversion Recall that the generalized power problem asks: Is there an algorithm that, on input u, v ∈ A∗,

determines whether or not there are nonzero integers k and l such that u k

=G v l, and

if so, outputs a solution pair (k, l) The main results of this chapter are as follows: Corollary 3.3.28 Let G = hA | Ri be a finitely generated δ-hyperbolic group with

A closed under inversion Let u, v ∈ A∗ be words representing elements of infinite

order in G Let β u and βv be the quasigeodesic rays based at 1 induced by u and v

respectively Then there is an algorithm that decides whether the pair of rays βu and

βv are divergent or non-divergent

Theorem 3.0.13 Let G = hA | Ri be a finitely generated δ-hyperbolic group with A

closed under inversion Then there is an algorithm deciding the generalized power

If g is an element of infinite order in a word hyperbolic group G = hAi, with A closed under inversion, and w ∈ A∗ is a geodesic representative of g, then there is a

geodesic path αw in the Cayley graph Γ(G,A) from 1 to g labeled by w It is known

that the ray in the Cayley graph formed by translating αw by positive powers of

g is a quasigeodesic The algorithms used to prove Theorem 3.0.13 and Theorem3.0.14 will make use of properties of quasigeodesics in δ-hyperbolic spaces Afterdefining δ-hyperbolic spaces and developing some basic properties of geodesics andquasigeodesics, we give some previously known algorithms to decide the word andorder problems in word hyperbolic groups, as well as an algorithm to decide the powerproblem in word hyperbolic groups (this problem can also be decided using methodsfound in [Lys]) Our proofs will make use of the fact that the pair of quasigeodesicrays induced by two words representing elements of infinite order in the Cayley graph

of a word hyperbolic group are either divergent or non-divergent Our algorithmswill be broken into subcases, depending on whether 0,1 or 2 of the words involved inthe membership question considered represent elements of infinite order In the casewhere both represent elements of infinite order, we show that one can detect whetherthe quasigeodesic rays induced by this pair of words are divergent or non-divergent

3.1 δ -hyperbolic Spaces and δ-hyperbolic Groups

The following definitions are taken from [BH] Chapter 3 (for another treatment ofthis material, see [A+]):

Definition 3.1.1 ([BH] p 12) Given a metric space X, a path c is a continuous mapping c : [a, b] → X We say that c joins the point c(a) to the point c(b) in X The

length of a path, l(c) := sup

a=t0≤t1≤ ≤t n =b

n−1

X

i=0

d(c(t i ), c(t i+1)), where the supremum is taken

over all possible partitions with a = t0 ≤ t1 ≤ ≤ t n = b and no bound on n A path

is rectifiable if for all t1,t2 satisfying a ≤ t1 < t2 ≤ b, the restriction of c mapping

[t1,t2] into X has finite length.

Notes:

(1) In everything that follows, all paths are assumed to be rectifiable In fact, wewill parameterize them so that each path is traversed in unit time and at constantspeed

(2) Eventually, X will be taken to be the Cayley graph of a group G with the path

metric Notice that in this case, the definition of a path given above corresponds tothe standard notion of a path in a Cayley graph

(3) By allowing either one or both of the endpoints of [a, b] to be infinite, one

can extend the definition of a path given above In the language of the definition

above, a ray is a continuous mapping c : [a, +∞) → X We say that the ray is based at the point c(a) ∈ X Similarly, a bi-infinite ray is a continuous mapping

c : (−∞, +∞) → X.

Definition 3.1.2 (see [A+] p 39) Let c be a rectifiable path in a metric space X For any pair of points x := c(t1), y := c(t2) in c, we define d c (x, y) := l(c′) where c′ is

the path from x to y when traveling along c.

Definition 3.1.3 ([BH] definition 1.3, p 4) Let c be a rectifiable path in a metric space X Let x := c(t1)and y := c(t2)be a pair of points in c If d c (x, y) = d(x, y) = |t1−t2|

for all t1,t2 ∈ [a, b], then c is a geodesic A metric space X is called geodesic if any pair of points in X can be joined by a geodesic path.

Let X be a geodesic metric space, and let δ > 0 be given A geodesic triangle

△(x, y, z) is a set of three points x, y, z ∈ X together with geodesic paths from x to y, from x to z, and from y to z A geodesic triangle △(x, y, z) is called δ-slim if each

δ δ

Figure 3.1: A δ-slim triangle

of its sides is contained in the δ-neighborhood of the union of the other two sides

A geodesic metric space X is δ-hyperbolic if every geodesic triangle is δ-slim (see

Figure 3.1)

Given any positive numbers a, b, c, there is a metric tree T (a, b, c) that has three vertices of valence one, one vertex of valence three, and edges of length a, b, c Such

a tree is called a tripod This definition can be extended to include degenerate

cases where one or more of a, b, c are zero In general, a tripod T (a, b, c) is a metric

simplicial tree with at most three edges and at most one vertex of degree greater thanone

Given any three points x, y, z in a metric space X, there are unique non-negative numbers a, b, c such that d(x, y) = a + b, d(x, z) = a + c and d(y, z) = b + c There

is an isometry from {x, y, z} to a subset of the vertices of T (a, b, c) (the vertices of valence one in the non-degenerate case) We label the vertices v x,v y,v z to reflect thecorrespondence induced by this isometry

Given a geodesic triangle, △ = △(x, y, z), let T△ := T (a, b, c), where a, b and c

are defined as above, and let o△ denote the central vertex of T△ Then the map

{x, y, z} → {v x,v y,v z} defined above extends uniquely to a map χ△ : △ → T△ whoserestriction to each side of △ is an isometry

Figure 3.2: A δ-thin triangle.

Definition 3.1.4 Let △ be a geodesic triangle in a metric space X and consider the

map χ△ : △ → T△ defined above Let δ > 0 △ is said to be δ-thin if p, q ∈ χ−1

△ (t)

implies d(p, q) ≤ δ, for all t ∈ T△ (see Figure 3.2)

Proposition 3.1.5 ([BH] Proposition 1.17 p 408) Let X be a geodesic metric space.

Then the following conditions are equivalent:

(1) There exists δ0> 0 such that every geodesic triangle in X is δ0-slim

(2) There exists δ1> 0 such that every geodesic triangle in X is δ1-thin

With this in mind, when we say that a space X is δ-hyperbolic, we assume that

δ has been chosen so that all geodesic triangles in X are both δ-thin and δ-slim We

will mainly use the second characterization of hyperbolicity from Proposition 3.1.5

Let G be a group with finite generating set A, with A closed under inversion Recall that the Cayley graph of G with respect to A, denoted Γ(G, A), can be made

into a geodesic metric space by taking each edge as isometric to the unit interval, andusing the path metric

Definition 3.1.6 A group G is said to be δ-hyperbolic with respect to a generating set A if Γ(G, A) with the path metric is a δ-hyperbolic metric space for some δ > 0.

3.2 Algorithms for δ-hyperbolic Groups

Recall: A group G with finite generating set A has decidable word problem (with respect to the generating set A) if there is an algorithm that, on input of an arbitrary word w ∈ (A ∪ A−1)∗, decides in a finite number of steps whether or not w = G 1

Definition 3.2.1 ([A+] 1.11 p 15) A Dehn presentation for a group G is a finite presentation hA | Ri such that any non-trivial word in F(A) that represents the identity element in G contains more than half of a prefix of some word in R That

is, if w ∈ F(A) is a reduced word and w = G 1, then there is a relator r ≡ uv ∈ R with l(u) > l(v) such that w ≡ w1uw2

Proposition 3.2.2 ([BH] 2.4, 2.5 pp 449-450) Let hA | Ri be a Dehn presentation for a group G Then the Word Problem for G with respect to A is decidable.

Proof: Let w0 ∈ (A ∪ A−1)∗ Without loss of generality, we may assume that our

Dehn Presentation includes relators of the form aa−1

= 1 and a−1a = 1 for a ∈ A If

w0 =G 1, then there is at least one r0 ∈ R satisfying r0 =G 1, r0 ≡ u0v0, |u0| > |v0|,

and u0 is a subword of w0 That is, w0 ≡ w′0u0w′′0 for some (possibly trivial) words

w′0,w′′0 ∈ (A ∪ A−1)∗ Let w1 ≡ w′

0v−10 w′′0 Then w0 =G w1 and |w0| > |w1| If w1 is

non-trivial, then the same procedure can be applied to w1 yielding a word w2 with

w2 =G w1, and |w1| > |w2| Repeatedly applying this process gives a sequence of words

satisfying |w0| > |w1| > |w k| Hence, for some k, |wk | = 0, thus w k ≡ 1

With this in mind, the following algorithm (Dehn’s algorithm) decides the Word

Problem in G with respect to A:

Step 1: Given hA | Ri, a Dehn presentation for a group G, construct U := {u ∈

(A ∪ A−1)∗ | there is some r ∈ R such that r ≡ uv with |u| > |v|}.

Step 2: For w ∈ (A ∪ A−1)∗, we check to see if any u ∈ U occurs as a subword of w so that w ≡ w′uw′′ If so we replace u with v, yielding a shorter word w1 ≡ w′vw′′.

Step 3: We repeat this process until there no u ∈ U occurs as a subword of the current word w k If w k ≡ 1, then output YES, otherwise, output NO 

Theorem 3.2.3 ([A+] Theorem 2.12, p 31) Let G = hAi be δ-hyperbolic, and let

R = {w ∈ F(A) | |w| ≤ 8δ and w = G 1} Then hA | Ri is a Dehn presentation for G Hence the word problem for G is decidable via Dehn’s Algorithm.

In fact, having a Dehn presentation completely characterizes hyperbolic groups:

Theorem 3.2.4 ([BH] Theorem 2.6, p 450 See also [Can], [Lys], or [A+]) A group

is hyperbolic if and only if it admits a finite Dehn presentation

A consequence of this characterization is that every word hyperbolic group G is finitely presented Moreover, if G is δ-hyperbolic with respect to a finite generating set A, and B is another finite generating set for G, then there is some δ′ > 0 such

that G is δ′-hyperbolic with respect to B Thus, hyperbolicity is independent of finite generating set If a group G is δ-hyperbolic with respect to some finite generating set

A, then we say that G is word hyperbolic.

Note: The fact that the word problem is decidable for a group G with respect to

a finite generating A, with A closed under inversion, allows us to compute distances between vertices in the Cayley graph Γ(G, A) To see this, notice that for any word

w ∈ (A ∪ A−1)∗of length n, by Proposition 2.1.14, we can find a geodesic representative

of w If bw is a geodesic representative of w, and |bw| = m, then the distance from 1 to

π(w) in Γ(G, A) is m.

3.2.2 The Order Problem

Recall that the order problem for a group G with generating set A, with A closed under inversion, asks the following: given a word w ∈ A∗, is there an algorithm that,

on input w, determines whether or not π(w) has finite order in G If π(w) has finite order, the algorithm outputs the order of π(w), and if π(w) has infinite order, the

algorithm stops after a finite number of steps and outputs ∞

The following Theorem and Corollary show that if G is word hyperbolic, then G

has decidable Order Problem (for another proof, see [Lys])

Theorem 3.2.5 ([Bra]) Let G be a δ-hyperbolic group with respect to a finite ating set A, with A closed under inversion Then every finite subgroup of G can be conjugated to lie within the (2δ + 1)-ball about the identity element in Γ(G, A).

gener-Corollary 3.2.6 ([Bra]) Let G be a hyperbolic group as above Then there are only finitely many conjugacy classes of finite subgroups of G, and every finite subgroup contains at most |A|2δ+1+ 1 elements

A consequence of this corollary is that in a δ-hyperbolic group G with finite erating set A, A closed under inversion, every g ∈ G with finite order has order

gen-≤ |A|2δ+1+ 1

Then the following algorithm decides the order problem in G:

Input: A word w ∈ A∗

Step 1: For k = 1 to |A|2δ+1 + 1, use Dehn’s Algorithm to check to see if w k =G 1 If

π(w) has finite order, this step finds the least k for which w k

=G1, which is the order

of π(w), and the algorithm stops and outputs this least k.

Step 2: If w k ,G 1 for 1 ≤ k ≤ |A|2δ+1+ 1, then the algorithm stops and outputs ∞

In light of the fact that there are algorithms to decide the word problem andthe order problem in hyperbolic groups, it is natural to wonder which of the other

generalizations of the word problem are decidable in the class of word hyperbolicgroups Perhaps we can decide membership in any rational subset, or at least infinitely generated submonoids or finitely generated subgroups of a word hyperbolicgroup Unfortunately (or perhaps fortunately, depending on your point of view) all

of these generalizations of the word problem end up being undecidable in the class ofword hyperbolic groups

This can be seen from following theorem, due to E Rips (see [Rip]):

Theorem 3.2.7 There is a word hyperbolic group G with finite generating set A such that the uniform generalized word problem for G is undecidable.

Rips actually shows a good deal more than this in his paper Since the uniformgeneralized word problem is undecidable in the class of word hyperbolic groups, weknow that the uniform submonoid membership problem and the rational membershipproblem are also undecidable, since an algorithm deciding either or these problems in

a group G could be used to decide membership in finitely generated subgroups of G.

Despite these negative results, one can still obtain some positive results for bership problems that sit between the word problem and the uniform generalized wordproblem These include the power problem and the double power problem The nextpart of this chapter will be dedicated to developing algorithms for these problems inthe class of word hyperbolic groups

mem-3.3 The Power Problem and the Generalized

Power Problem in Word Hyperbolic Groups

Recall that the generalized power problem asks: given a finitely generated group

G = hAi for some finite set A that is closed under inversion, is there an algorithm