SOME DECISION PROBLEMS IN GROUP THEORY by Justin A. James A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors Susan M. Hermiller and John C. Meakin Lincoln, Nebraska May, 2006 UMI Number: 3208081 3208081 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. SOME DECISION PROBLEMS IN GROUP THEORY 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 hyperbolic groups. Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro, and Short showed that elements of infinite order in word hyperbolic groups induce quasigeodesic rays in the 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 we give an algorithm detecting which option occurs for a given pair of quasigeodesic rays. Since solutions to the generalized power problem correspond to points of intersection along these rays, this decides instances of the generalized power problem involving elements of infinite order. For finite order instances, we use the algorithm deciding the word problem in word hyperbolic groups finitely many times. We extend this result to obtain an algorithm deciding membership in the product of two cyclic submonoids of a word hyperbolic group. We also give an algorithm deciding membership in finitely generated submonoids of the free product of two finitely presented groups, provided there 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 generalized word problem is decidable in the free product of two groups if it is decidable in each factor. Since rational membership is known to be decidable for free groups, free abelian groups, virtually free groups, and virtually free abelian groups, our algorithm can be used to decide membership in finitely generated submonoids of a free product of groups with factors drawn from these classes of groups. ACKNOWLEDGEMENTS 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 and for 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 into reading the drafts leading up to this final document. I would also like to thank the other members of my supervisory committee: Dr. Mark Brittenham, Dr. Allan Donsig, and Dr. David Swanson. I would particularly like to thank Dr. Brittenham for his suggestions which improved the clarity of my proofs. I would also like to thank Dr. Brittenham and Dr. Donsig for reading the penultimate draft and providing many helpful comments. I am especially grateful for the camaraderie among 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 their friendship. You have enriched my life in innumerable ways. I would be remiss if I failed to thank my friends in the UNL Navigators and at Oak Lake Evangelical Free Church. Without your prayers and support, I would never have made it to this point. 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 mean more to me than words are able to express. Lastly, I dedicate this dissertation to the 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 and hidden things; he knows what is in the darkness, and the light dwells with him.” Daniel 2:20b-22 (ESV) Contents 1 Introduction 1 2 Preliminaries 7 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 4.2.1 The Automaton Construction . . . . . . . . . . . . . . . . . . 76 4.2.2 An Algorithm to Recognize Short Elements . . . . . . . . . . 81 4.2.3 Generating Set Completion . . . . . . . . . . . . . . . . . . . 93 Bibliography 104 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 p oints an equal distance along a pair of geodesic rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.9 The case where p = x 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.10 The case where p ∈ (x 0 , o 2 ] . . . . . . . . . . . . . . . . . . . . . . . . 46 3.11 The case where p ∈ (o 2 , y 2 ) . . . . . . . . . . . . . . . . . . . . . . . . 46 3.12 The case where p ∈ [y 2 , ∞) . . . . . . . . . . . . . . . . . . . . . . . . 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 b etween 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 t 2 . . . . . . . . . . . . . . . . . . . . . . . 54 3.19 A pair of vertices on non-divergent quasigeodesic rays at a distance of < K from one another. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.20 A second pair of vertices on non-divergent quasigeodesic rays at a dis- 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,C 2 ,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 v 1 . . . . . . . . . . . . . . . . . . 86 4.7 The case where x 1 is a syllable of  u 1 . . . . . . . . . . . . . . . . . . . 87 4.8 The case where x 1 is a syllable of  u 2 . . . . . . . . . . . . . . . . . . . 87 4.9 The case where x 1 is a syllable of  u 3 . . . . . . . . . . . . . . . . . . . 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  u i s and there is a cancellation between them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.12 The case where x l+1 is in  u t+2 . . . . . . . . . . . . . . . . . . . . . . . 89 4.13 The path in the automaton from ǫ to the final state S 1 , labeled by v 1 . 90 4.14 The cancellations forming v 2 when y 1 is a segment of  u m 1 , and v 2 = G y 1 . 91 4.15 The cancellations forming v 2 when y 1 is a segment of  u m 1 and v 2  G y 1 . 91 4.16 The cancellations forming v 2 when y 1 is not a segment of  u m 1 and v 2  G y 1 . 91 4.17 The case where the last surviving segment is in  u N . . . . . . . . . . . 93 4.18 The case where the last surviving segment is in  u m 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  u j+1  u j+k 1 −1 = G T −1 1 R −1 k 1 . . . . . . 98 1 Chapter 1 Introduction In 1911, Dehn formulated three fundamental decision problems for groups: the word problem, the conjugacy problem, and t he 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 that there is a finitely generated group with undecidable word problem. Since then, there has been much progress toward the goal of determining which classes of groups have decidable word problem and which have undecidable word problem, but many open questions remain. If we restrict our attention to finitely generated groups with decidable word prob- lem, there are several generalizations of the word problem that can be considered. These include: deciding membership in finitely generated subgroups (the uniform generalized word problem), deciding membership in finitely generated submonoids 2 (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 (th e 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 these decision problems, there is an example of a finitely generated group for which the word problem is decidable, but the given problem is undecidable (references for these examples 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 these problems 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, Zeph Grunschlag [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 gener- ated free abelian groups as well. This dissertation explores the decidability of the aforementioned generalizations of the word problem. The main results are as follows: Corollary 3.3.28 Let G = A | R 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 β v 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 β u and β v are divergent or non-divergent. [...]... taken to be isometric to the unit interval [0, 1] in R The distance between arbitrary points in the graph (including points in the interior of edges) is defined to be the length of the shortest path in the graph 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... 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 . Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. SOME DECISION PROBLEMS IN GROUP THEORY Justin. hyperbolic group. In Chapter 4, we shift our focus to another perspective on decision problems in groups. Rather than considering a class of groups, we consider the effect of classical group theoretic. this decides instances of the generalized power problem involving elements of in nite order. For finite order instances, we use the algorithm deciding the word problem in word hyperbolic groups finitely