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Undecidability of the Word Problem for Groups: the Point of View of Rewriting Theory
Universita` degli Studi Roma Tre Facolta` di Scienze M.F.N. Corso di Laurea in Matematica Tesi di Laurea in Matematica Undecidability of the word problem for groups: the point of view of rewriting theory Candidato Relatori Matteo Acclavio Prof. Y. Lafont ..................................... Prof. L. Tortora de falco ...................................... questa tesi ´estata redatta nell'ambito del Curriculum Binazionale di Laurea Magistrale in Logica, con il sostegno dell'Universit´aItalo-Francese (programma Vinci 2009) Anno Accademico 2011-2012 Ottobre 2012 Classificazione AMS: Parole chiave: \There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, . " Italian nursery rhyme Even if you don't know this tale, it's easy to understand that this could continue indefinitely, but it doesn't have to. If now we want to know if the nar- ration will finish, this question is what is called an undecidable problem: we'll need to listen the tale until it will finish, but even if it will not, one can never say it won't stop since it could finish later. those things make some people loose sleep, but usually children, bored, fall asleep. More precisely a decision problem is given by a question regarding some data that admit a negative or positive answer, for example: \is the integer number n odd?" or \ does the story of the king on the sofa admit an happy ending?". -
CLASS DESCRIPTIONS—WEEK 2, MATHCAMP 2016 Contents 9:10
CLASS DESCRIPTIONS|WEEK 2, MATHCAMP 2016 Contents 9:10 Classes 2 Dynamical Systems 2 Field Extensions and Galois Theory (Week 1 of 2) 2 Model Theory 2 Neural Networks 3 Problem Solving: Induction 3 10:10 Classes 4 Almost Planar 4 Extending Inclusion-Exclusion 4 Multilinear Algebra 5 The Word Problem for Groups 5 Why Are We Learning This? A seminar on the history of math education in the U.S. 6 11:10 Classes 6 Building Mathematical Structures 6 Graph Minors 7 History of Math 7 K-Theory 7 Linear Algebra (Week 2 of 2) 7 The Banach{Tarski Paradox 8 1:10 Classes 8 Divergent Series 8 Geometric Group Theory 8 Group Actions 8 The Chip-Firing Game 9 Colloquia 9 There Are Eight Flavors of Three-Dimensional Geometry 9 Tropical Geometry 9 Oops! I Ran Out of Axioms 10 Visitor Bios 10 Moon Duchin 10 Sam Payne 10 Steve Schweber 10 1 MC2016 ◦ W2 ◦ Classes 2 9:10 Classes Dynamical Systems. ( , Jane, Tuesday{Saturday) Dynamical systems are spaces that evolve over time. Some examples are the movement of the planets, the bouncing of a ball around a billiard table, or the change in the population of rabbits from year to year. The study of dynamical systems is the study of how these spaces evolve, their long-term behavior, and how to predict the future of these systems. It is a subject that has many applications in the real world and also in other branches of mathematics. In this class, we'll survey a variety of topics in dynamical systems, exploring both what is known and what is still open. -
BRANCHING PROGRAM UNIFORMIZATION, REWRITING LOWER BOUNDS, and GEOMETRIC GROUP THEORY IZAAK MECKLER Mathematics Department U.C. B
BRANCHING PROGRAM UNIFORMIZATION, REWRITING LOWER BOUNDS, AND GEOMETRIC GROUP THEORY IZAAK MECKLER Mathematics Department U.C. Berkeley Berkeley, CA Abstract. Geometric group theory is the study of the relationship between the algebraic, geo- metric, and combinatorial properties of finitely generated groups. Here, we add to the dictionary of correspondences between geometric group theory and computational complexity. We then use these correspondences to establish limitations on certain models of computation. In particular, we establish a connection between read-once oblivious branching programs and growth of groups. We then use Gromov’s theorem on groups of polynomial growth to give a simple argument that if the word problem of a group G is computed by a non-uniform family of read-once, oblivious, polynomial-width branching programs, then it is computed by an O(n)-time uniform algorithm. That is, efficient non-uniform read-once, oblivious branching programs confer essentially no advantage over uniform algorithms for word problems of groups. We also construct a group EffCirc which faithfully encodes reversible circuits and note the cor- respondence between certain proof systems for proving equations of circuits and presentations of groups containing EffCirc. We use this correspondence to establish a quadratic lower bound on the proof complexity of such systems, using geometric techniques which to our knowledge are new to complexity theory. The technical heart of this argument is a strengthening of the now classical the- orem of geometric group theory that groups with linear Dehn function are hyperbolic. The proof also illuminates a relationship between the notion of quasi-isometry and models of computation that efficiently simulate each other. -
![[Math.GR] 2 Mar 2002 ∆ Missing”](https://docslib.b-cdn.net/cover/1749/math-gr-2-mar-2002-missing-1051749.webp)
[Math.GR] 2 Mar 2002 ∆ Missing”
COXETER GROUPS, 2-COMPLETION, PERIMETER REDUCTION AND SUBGROUP SEPARABILITY Paul E. Schupp Abstract. We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index. §1. Introduction. Several years ago the author [S2] raised the question of whether or not small cancellation methods could be used to investigate questions about finitely gener- ated subgroups of “sufficiently nice” groups— in particular the solvability of the membership problem and the Howson property. Rips [R] replied “No” by con- ′ 1 structing, for every metric small cancellation condition C n , finitely presented groups which satisfy the condition but which have unsolvable membership prob- lem and are neither Howson nor coherent. However, recent work of McCammond and Wise [M-W] and of Arzhantseva and Olshanskii[A-O] shows that the answer is actually “Yes” in many cases. McCammond and Wise introduced the use of “distributive” small cancellation hypotheses where the condition involves how the generators are distributed among all the defining relators and the ingenious idea of “perimeter reduction” where one counts “what is missing”. In this paper we introduce the idea that if one has a “suitable” subgroup graph, ∆1(H), of a subgroup H, one can construct from it a “complete” subgroup graph, arXiv:math/0203020v1 [math.GR] 2 Mar 2002 ∆2(H) which has most of the properties of subgroup graphs in the case of free groups and which directly reveals desired information about H. -
The Compressed Word Problem for Groups
Markus Lohrey The Compressed Word Problem for Groups – Monograph – April 3, 2014 Springer Dedicated to Lynda, Andrew, and Ewan. Preface The study of computational problems in combinatorial group theory goes back more than 100 years. In a seminal paper from 1911, Max Dehn posed three decision prob- lems [46]: The word problem (called Identitatsproblem¨ by Dehn), the conjugacy problem (called Transformationsproblem by Dehn), and the isomorphism problem. The first two problems assume a finitely generated group G (although Dehn in his paper requires a finitely presented group). For the word problem, the input con- sists of a finite sequence w of generators of G (also known as a finite word), and the goal is to check whether w represents the identity element of G. For the con- jugacy problem, the input consists of two finite words u and v over the generators and the question is whether the group elements represented by u and v are conju- gated. Finally, the isomorphism problem asks, whether two given finitely presented groups are isomorphic.1 Dehns motivation for studying these abstract group theo- retical problems came from topology. In a previous paper from 1910, Dehn studied the problem of deciding whether two knots are equivalent [45], and he realized that this problem is a special case of the isomorphism problem (whereas the question of whether a given knot can be unknotted is a special case of the word problem). In his paper from 1912 [47], Dehn gave an algorithm that solves the word problem for fundamental groups of orientable closed 2-dimensional manifolds. -
Séminaire BOURBAKI Octobre 2018 71E Année, 2018–2019, N 1154 ESPACES ET GROUPES NON EXACTS ADMETTANT UN PLONGEMENT GROSSIER
S´eminaireBOURBAKI Octobre 2018 71e ann´ee,2018{2019, no 1154 ESPACES ET GROUPES NON EXACTS ADMETTANT UN PLONGEMENT GROSSIER DANS UN ESPACE DE HILBERT [d'apr`esArzhantseva, Guentner, Osajda, Spakula]ˇ by Ana KHUKHRO 1. INTRODUCTION The landscape of modern group theory has been shaped by the use of geometry as a tool for studying groups in various ways. The concept of group actions has always been at the heart of the theory, and the idea that one can link the geometry of the space on which the group acts to properties of the group, or that one can view groups as geometric objects themselves, has opened up many possibilities of interaction between algebra and geometry. Given a finitely generated discrete group and a finite generating set of this group, one can construct an associated graph called a Cayley graph. This graph has the set of elements of the group as its vertex set, and two vertices are connected by an edge if one can obtain one from the other by multiplying on the right by an element of the generating set. This gives us a graph on which the group acts by isometries, viewing the graph as a metric space with the shortest path metric. While a different choice of generating set will result in a non-isomorphic graph, the two Cayley graphs will be the same up to quasi-isometry, a coarse notion of equivalence for metric spaces. The study of groups from a geometric viewpoint, geometric group theory, often makes use of large-scale geometric information. -
Amenable Groups Without Finitely Presented Amenable Covers
Bull. Math. Sci. (2013) 3:73–131 DOI 10.1007/s13373-013-0031-5 Amenable groups without finitely presented amenable covers Mustafa Gökhan Benli · Rostislav Grigorchuk · Pierre de la Harpe Received: 10 June 2012 / Revised: 25 June 2012 / Accepted: 2 February 2013 / Published online: 1 May 2013 © The Author(s) 2013. This article is published with open access at SpringerLink.com Abstract The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups G with the following properties: (i) G is finitely generated, (ii) G is amenable, e.g. of intermediate growth, (iii) any finitely presented group with a quotient isomorphic to G contains non-abelian free subgroups, or the stronger (iii’) any finitely presented group with a quotient isomorphic to G is large. Keywords Finitely generated groups · Finitely presented covers · Self-similar groups · Metabelian groups · Soluble groups · Groups of intermediate growth · Amenable groups · Groups with non-abelian free subgroups · Large groups Mathematics Subject Classification (2000) 20E08 · 20E22 · 20F05 · 43A07 Communicated by Dr. Efim Zelmanov. We are grateful to the Swiss National Science Foundation for supporting a visit of the first author to Geneva. The first author is grateful to the department of mathematics, University of Geneva, for their hospitality during his stay. The first and the second authors are partially supported by NSF grant DMS – 1207699. The second author was supported by ERC starting grant GA 257110 “RaWG”. M. G. Benli (B) · R. Grigorchuk Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843–3368, USA e-mail: [email protected] R. -
Some Characterizations of Howson PC-Groups
AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMATIQUES` Some characterizations of Howson PC-groups ∗Jordi Delgado Resum (CAT) Universitat Polit`ecnica de Es demostra que, a la classe dels grups parcialment commutatius, les condicions Catalunya d'´esserHowson, ´essertotalment residualment lliure, i ´esserproducte lliure de grups [email protected] lliure-abelians, s´onequivalents. ∗Corresponding author Abstract (ENG) We show that, within the class of partially commutative groups, the conditions of being Howson, being fully residually free, and being a free product of free-abelian groups, are equivalent. Acknowledgement The author gratefully acknowledges the Keywords: PC-group, residually free, support of Universitat Polit`ecnica de Howson group. Catalunya through PhD grant number MSC (2010): 20Exx. 81--727, and the partial support from the Received: May 9, 2014. MEC (Spanish Government) through re- Accepted: May 20, 2014. search project number MTM2011-25955. http://reportsascm.iec.cat Reports@SCM 1 (2014), 33{38; DOI:10.2436/20.2002.02.3. 33 Some characterizations of Howson PC-groups In [10], the authors study the family of finitely generated partially commutative groups for which the fixed points subgroup of every endomorphism is finitely generated. Concretely, they characterize this family as those groups consisting in (finite) free products of finitely generated free-abelian groups. In this note we provide an elementary proof for two extra characterizations of this same family, namely: being Howson, and being a limit group. Moreover, we observe that, for some of the properties, no restriction in the cardinal of the generating set is needed, and the result holds in full generality (i.e. -
Kurosh Rank of Intersections of Subgroups of Free Products of Right-Orderable Groups Yago Antol´In, Armando Martino and Inga Schwabrow
Math. Res. Lett. c 2014 International Press Volume 21, Number 04, 649–661, 2014 Kurosh rank of intersections of subgroups of free products of right-orderable groups Yago Antol´ın, Armando Martino and Inga Schwabrow We prove that the reduced Kurosh rank of the intersection of two subgroups H and K of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of H and K. In particular, taking the fundamental group of a graph of groups with trivial vertex and edge groups, and its Bass–Serre tree, our theorem becomes the desired inequality of the usual strengthened Hanna Neumann conjecture for free groups. 1. Introduction Let H and K be subgroups of a free group F, and r(H)=max{0, rank(H) − 1}. It was an open problem dating back to the 1950s to find an optimal bound of the rank of H ∩ K in terms of the ranks of H and K. In [14], Hanna Neumann proved the following: (1) r(H ∩ K) ≤ 2 · r(H) · r(K). The Hanna Neumann conjecture says that (1) holds replacing the 2 with a 1. Later, in [15], Walter Neumann improved (1) to (2) r(Hg ∩ K) ≤ 2 · r(H) · r(K), g∈K\F/H where Hg = g−1Hg. The strengthened Hanna Neumann conjecture,intro- duced by Walter Neumann, says that (2) holds replacing the 2 with a 1. These two conjectures have received a lot of attention, and recently, Igor Mineyev [13] proved that both conjectures are true (independently and at the same time, Joel Friedman also proved these conjectures (see [5, 8])). -
Formal Languages and the Word Problem for Groups 1 Introduction 2
Formal Languages and the Word Problem for Groups Iain A. Stewart∗ and Richard M. Thomas† Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, England. 1 Introduction The aim of this article is to survey some connections between formal language theory and group theory with particular emphasis on the word problem for groups and the consequence on the algebraic structure of a group of its word problem belonging to a certain class of formal languages. We define our terms in Section 2 and then consider the structure of groups whose word problem is regular or context-free in Section 3. In Section 4 we look at groups whose word- problem is a one-counter language, and we move up the Chomsky hierarchy to briefly consider what happens above context-free in Section 5. In Section 6, we see what happens if we consider languages lying in certain complexity classes, and we summarize the situation in Section 7. For general background material on group theory we refer the reader to [25, 26], and for formal language theory to [7, 16, 20]. 2 Word problems and decidability In this section we set up the basic notation we shall be using and introduce the notions of “word problems” and “decidability”. In order to consider the word problem of a group as a formal language, we need to introduce some terminology. Throughout, if Σ is a finite set (normally referred to in this context as an alphabet) then we let Σ∗ denote the set of all finite words of symbols from Σ, including the empty word , and Σ+ denote the set of all non-empty finite words of symbols from Σ. -
![[Math.GR] 31 Dec 2002 Upre Ne S Rn O DMS-9971511](https://docslib.b-cdn.net/cover/9593/math-gr-31-dec-2002-upre-ne-s-rn-o-dms-9971511-2979593.webp)
[Math.GR] 31 Dec 2002 Upre Ne S Rn O DMS-9971511
COHERENCE, LOCAL QUASICONVEXITY, AND THE PERIMETER OF 2-COMPLEXES JONATHAN P. MCCAMMOND 1 AND DANIEL T. WISE 2 Abstract. A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence and local quasiconvexity of many small cancellation groups. Contents 1. Introduction 2 2. Perimeter 5 3. Coherence theorem 12 4. Attachments 15 5. 2-cell coherence theorem 19 6. Algorithms 21 7. Path coherence theorem 24 8. One-relator groups with torsion 30 9. Small cancellation I 32 arXiv:math/0212381v1 [math.GR] 31 Dec 2002 10. Fan coherence theorems 36 11. Quasi-isometries and quasiconvexity 40 12. Fan quasiconvexity theorems 44 13. Small cancellation II 48 14. 3-manifold groups 51 15. Related properties 53 1Supported under NSF grant no. DMS-99781628 2Supported as an NSF Postdoctoral Fellow under grant no. -
Fifty Years of Homotopy Theory
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number 1, January 1983 FIFTY YEARS OF HOMOTOPY THEORY BY GEORGE W. WHITEHEAD The subject of homotopy theory may be said to have begun in 1930 with the discovery of the Hopf map. Since I began to work under Norman Steenrod as a graduate student at Chicago in 1939 and received my Ph.D. in 1941, I have been active in the field for all but the first ten years of its existence. Thus the present account of the development of the subject is based, to a large extent, on my own recollections. I have divided my discussion into two parts, the first covering the period from 1930 to about 1960 and the second from 1960 to the present. Each part is accompanied by a diagram showing the connections among the results dis cussed, and one reason for the twofold division is the complication of the diagram that would result were we to attempt to merge the two eras into one. The dating given in this paper reflects, not the publication dates of the papers involved, but, as nearly as I can determine them, the actual dates of discovery. In many cases, this is based on my own memory; this failing, I have used the date of the earliest announcement in print of the result (for example, as the abstract of a paper presented to the American Mathematical Society or as a note in the Comptes Rendus or the Proceedings of the National Academy). Failing these, I have used the date of submission of the paper, whenever available.