From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups
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Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a -
On Dehn Functions and Products of Groups
TRANSACTIONSof the AMERICANMATHEMATICAL SOCIETY Volume 335, Number 1, January 1993 ON DEHN FUNCTIONS AND PRODUCTS OF GROUPS STEPHEN G. BRICK Abstract. If G is a finitely presented group then its Dehn function—or its isoperimetric inequality—is of interest. For example, G satisfies a linear isoperi- metric inequality iff G is negatively curved (or hyperbolic in the sense of Gro- mov). Also, if G possesses an automatic structure then G satisfies a quadratic isoperimetric inequality. We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and HNN extensions. 0. Introduction The study of isoperimetric inequalities for finitely presented groups can be approached in two different ways. There is the geometric approach (see [Gr]). Given a finitely presented group G, choose a compact Riemannian manifold M with fundamental group being G. Then consider embedded circles which bound disks in M, and search for a relationship between the length of the circle and the area of a minimal spanning disk. One can then triangulate M and take simplicial approximations, resulting in immersed disks. What was their area then becomes the number of two-cells in the image of the immersion counted with multiplicity. We are thus led to a combinatorial approach to the isoperimetric inequality (also see [Ge and CEHLPT]). We start by defining the Dehn function of a finite two-complex. Let K be a finite two-complex. If w is a circuit in K^, null-homotopic in K, then there is a Van Kampen diagram for w , i.e. -
Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Introduction to knots and embeddings 13 Lecture 3: Link diagrams and Alexander polynomial skein relations 17 Lecture 4: Handle decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Morse functions from handle decompositions 24 Lecture 6: Handle-bodies and Heegaard diagrams 28 Lecture 7: Fundamental group presentations from Heegaard diagrams 36 Lecture 8: Alexander polynomials from fundamental groups 39 Lecture 9: Fox calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston classification for mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn surgery 64 Lecture 15: 3-manifolds from Dehn surgery 68 Lecture 16: Seifert fibered spaces 72 Lecture 17: Hyperbolic manifolds 76 Lecture 18: Embedded surface representatives 80 Lecture 19: Incompressible and essential surfaces 83 Lecture 20: Connected sum 86 Lecture 21: JSJ decomposition and geometrization 89 Lecture 22: Turaev torsion and knot decompositions 92 Lecture 23: Foliations 96 Lecture 24. Taut Foliations 98 Errata: Catalogue of errors/changes/addenda 102 References 106 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . ◦ ν(X) { the interior of ν(X). This notation is somewhat redundant, but emphasises openness. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Knot Theory and the Alexander Polynomial
Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ............................... -
Cubulating Random Groups at Density Less Than 1/6
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 9, September 2011, Pages 4701–4733 S 0002-9947(2011)05197-4 Article electronically published on March 28, 2011 CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6 YANN OLLIVIER AND DANIEL T. WISE Abstract. 1 We prove that random groups at density less than 6 act freely and cocompactly on CAT(0) cube complexes, and that random groups at density 1 less than 5 have codimension-1 subgroups. In particular, Property (T ) fails 1 to hold at density less than 5 . Abstract. Nous prouvons que les groupes al´eatoires en densit´e strictement 1 inf´erieure `a 6 agissent librement et cocompactement sur un complexe cubique 1 CAT(0). De plus en densit´e strictement inf´erieure `a 5 , ils ont un sous-groupe de codimension 1; en particulier, la propri´et´e(T )n’estpasv´erifi´ee. Introduction Gromov introduced in [Gro93] the notion of a random finitely presented group on m 2 generators at density d ∈ (0; 1). The idea is to fix a set {g1,...,gm} of generators and to consider presentations with (2m − 1)d relations each of which is a random reduced word of length (Definition 1.1). The density d is a measure of the size of the number of relations as compared to the total number of available relations. See Section 1 for precise definitions and basic properties, and see [Oll05b, Gro93, Ghy04, Oll04] for a general discussion on random groups and the density model. One of the striking facts Gromov proved is that a random finitely presented 1 {± } 1 group is infinite, hyperbolic at density < 2 , and is trivial or 1 at density > 2 , with probability tending to 1 as →∞. -
On a Small Cancellation Theorem of Gromov
On a small cancellation theorem of Gromov Yann Ollivier Abstract We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from labelled graphs. In this paper we present a combinatorial proof of a small cancellation theorem stated by M. Gromov in [Gro03], which strongly generalizes the usual tool of small cancellation. Our aim is to complete the six-line-long proof given in [Gro03] (which invokes geometric arguments). Small cancellation theory is an easy-to-apply tool of combinatorial group theory (see [Sch73] for an old but nicely written introduction, or [GH90] and [LS77]). In one of its forms, it basically asserts that if we face a group presentation in which no two relators share a common subword of length greater than 1/6 of their length, then the group so defined is hyperbolic (in the sense of [Gro87], see also [GH90] or [Sho91] for basic properties), and infinite except for some trivial cases. The theorem extends these conclusions to much more general situations. Sup- pose that we are given a finite graph whose edges are labelled by generators of the free group Fm and their inverses (in a reduced way, see definition below). If no word of length greater than 1/6 times the length of the smallest loop of the graph appears twice on the graph, then the presentation obtained by taking as relations all the words read on all loops of the graph defines a hyperbolic group which (if the rank of the graph is at least m +1, to avoid trivial cases) is infinite. -
Case for Support
Solving word problems via generalisations of small cancellation ATrackRecord A.1 The Research Team Prof. Stephen Linton is a Professor of Computer Science at the University of St Andrews. He has worked in computational algebra since 1986 and has coordinated the development of the GAP system [6] since its transfer from Aachen in 1997 (more recently, in cooperation with threeothercentres).HehasbeenDirectoroftheCentre for Interdisciplinary Research in Computational Algebra (CIRCA) since its inception in 2000. His work includes new algorithms for algebraic problems, most relevantly workongeneralisationsoftheTodd-Coxeteralgorithm, novel techniques for combining these algorithms into the GAP system, and their application to research problems in mathematics and computer science. He is an editor of the international journal Applicable Algebra and Error Correcting Codes.HehasbeenprincipalinvestigatoronfivelargeEPSRCgrants, including EP/C523229 “Multidisciplinary Critical Mass in Computational AlgebraandApplications”(£1.1m) and EP/G055181 “High Performance Computational Algebra” (£1.5m across four sites). He is the Coordinator of the EU FP6 SCIEnce project (RII3-CT-026133, 3.2m Euros), developing symboliccomputationsoftwareasresearchinfrastructure. He won Best Paper award at the prestigious IEEE Visualisationconferencein2003forapaperapplying Computational Group Theory to the analysis of a family of algorithms in computer graphics. Dr. Max Neunh¨offer is currently a Senior Research Fellow in the School of Mathematics and Statistics at St Andrews, working on EP/C523229 “Multidisciplinary Critical Mass in Computational Algebra”, and will advance to a Lectureship in Mathematics in September 2010. HehasbeenatStAndrewssince2007,and before that worked for 10 years at Lehrstuhl D f¨ur MathematikatRWTHAachen,Germany.Hehasbeen involved in the development of GAP since the 1990s, both developing the core system and writing package code. -
Max Dehn: His Life, Work, and Influence
Mathematisches Forschungsinstitut Oberwolfach Report No. 59/2016 DOI: 10.4171/OWR/2016/59 Mini-Workshop: Max Dehn: his Life, Work, and Influence Organised by David Peifer, Asheville Volker Remmert, Wuppertal David E. Rowe, Mainz Marjorie Senechal, Northampton 18 December – 23 December 2016 Abstract. This mini-workshop is part of a long-term project that aims to produce a book documenting Max Dehn’s singular life and career. The meet- ing brought together scholars with various kinds of expertise, several of whom gave talks on topics for this book. During the week a number of new ideas were discussed and a plan developed for organizing the work. A proposal for the volume is now in preparation and will be submitted to one or more publishers during the summer of 2017. Mathematics Subject Classification (2010): 01A55, 01A60, 01A70. Introduction by the Organisers This mini-workshop on Max Dehn was a multi-disciplinary event that brought together mathematicians and cultural historians to plan a book documenting Max Dehn’s singular life and career. This long-term project requires the expertise and insights of a broad array of authors. The four organisers planned the mini- workshop during a one-week RIP meeting at MFO the year before. Max Dehn’s name is known to mathematicians today mostly as an adjective (Dehn surgery, Dehn invariants, etc). Beyond that he is also remembered as the first mathematician to solve one of Hilbert’s famous problems (the third) as well as for pioneering work in the new field of combinatorial topology. A number of Dehn’s contributions to foundations of geometry and topology were discussed at the meeting, partly drawing on drafts of chapters contributed by John Stillwell and Stefan M¨uller-Stach, who unfortunately were unable to attend. -
Lectures on the Mapping Class Group of a Surface
LECTURES ON THE MAPPING CLASS GROUP OF A SURFACE THOMAS KWOK-KEUNG AU, FENG LUO, AND TIAN YANG Abstract. In these lectures, we give the proofs of two basic theorems on surface topology, namely, the work of Dehn and Lickorish on generating the mapping class group of a surface by Dehn-twists; and the work of Dehn and Nielsen on relating self-homeomorphisms of a surface and automorphisms of the fundamental group of the surface. Some of the basic materials on hyper- bolic geometry and large scale geometry are introduced. Contents Introduction 1 1. Mapping Class Group 2 2. Dehn-Lickorish Theorem 13 3. Hyperbolic Plane and Hyperbolic Surfaces 22 3.1. A Crash Introduction to the Hyperbolic Plane 22 3.2. Hyperbolic Geometry on Surfaces 29 4. Quasi-Isometry and Large Scale Geometry 36 5. Dehn-Nielsen Theorem 44 5.1. Injectivity of ª 45 5.2. Surjectivity of ª 46 References 52 2010 Mathematics Subject Classi¯cation. Primary: 57N05; Secondary: 57M60, 57S05. Key words and phrases. Mapping class group, Dehn-Lickorish, Dehn-Nielsen. i ii LECTURES ON MAPPING CLASS GROUPS 1 Introduction The purpose of this paper is to give a quick introduction to the mapping class group of a surface. We will prove two main theorems in the theory, namely, the theorem of Dehn-Lickorish that the mapping class group is generated by Dehn twists and the theorem of Dehn-Nielsen that the mapping class group is equal to the outer-automorphism group of the fundamental group. We will present a proof of Dehn-Nielsen realization theorem following the argument of B. -
Arxiv:1911.02470V3 [Math.GT] 20 Apr 2021 Plicial Volume of Gr As Kgrk := Kαr,Rk1, the L -Semi-Norm of the Fundamental Class Αr (Section 3.1)
Simplicial volume of one-relator groups and stable commutator length Nicolaus Heuer, Clara L¨oh April 21, 2021 Abstract A one-relator group is a group Gr that admits a presentation hS j ri with a single relation r. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If r 2 F (S)0, we introduce a simplicial volume kGrk for one-relator groups. We relate this invariant to the stable commutator length sclS (r) of the element r 2 F (S). We show that often (though not always) the linear relationship kGrk = 4·sclS (r)−2 holds and that every rational number modulo 1 is the simplicial volume of a one-relator group. Moreover, we show that this relationship holds approximately for proper powers and for elements satisfying the small cancellation condition C0(1=N), with a multiplicative error of O(1=N). This allows us to prove for random 0 elements of F (S) of length n that kGrk is 2 log(2jSj − 1)=3 · n= log(n) + o(n= log(n)) with high probability, using an analogous result of Calegari{ Walker for stable commutator length. 1 Introduction A one-relator group is a group Gr that admits a presentation hS j ri with a single relation r 2 F (S). This rich and well studied class of groups in Geometric Group Theory generalises surface groups and shares many properties with them. A common theme is to relate the geometric properties of a classifying space of Gr to the algebraic properties of the relator r 2 F (S). -
Negative Curvature in Graphical Small Cancellation Groups
NEGATIVE CURVATURE IN GRAPHICAL SMALL CANCELLATION GROUPS GOULNARA N. ARZHANTSEVA, CHRISTOPHER H. CASHEN, DOMINIK GRUBER, AND DAVID HUME Abstract. We use the interplay between combinatorial and coarse geometric versions 0 of negative curvature to investigate the geometry of infinitely presented graphical Gr (1=6) small cancellation groups. In particular, we characterize their `contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group G containing an element g that is strongly contracting with respect to one finite generating set of G 0 and not strongly contracting with respect to another. In the case of classical C (1=6) small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. 0 We show that many graphical Gr (1=6) small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. 0 In the course of our analysis we show that if the defining graph of a graphical Gr (1=6) small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero. 1. Introduction Graphical small cancellation theory was introduced by Gromov as a powerful tool for constructing finitely generated groups with desired geometric and analytic properties [23].