Solvable Groups with Polynomial Dehn Functions
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 the Dehn Functions of K\" Ahler Groups
ON THE DEHN FUNCTIONS OF KAHLER¨ GROUPS CLAUDIO LLOSA ISENRICH AND ROMAIN TESSERA Abstract. We address the problem of which functions can arise as Dehn functions of K¨ahler groups. We explain why there are examples of K¨ahler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a K¨ahler group which has Dehn function bounded below by a cubic function 6 and above by n . As a consequence we obtain that for a compact K¨ahler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function. 1. Introduction A K¨ahler group is a group which can be realized as fundamental group of a compact K¨ahler manifold. K¨ahler groups form an intriguing class of groups. A fundamental problem in the field is Serre’s question of “which” finitely presented groups are K¨ahler. While on one side there is a variety of constraints on K¨ahler groups, many of them originating in Hodge theory and, more generally, the theory of harmonic maps on K¨ahler manifolds, examples have been constructed that show that the class is far from trivial. Filling the space between examples and constraints turns out to be a very hard problem. This is at least in part due to the fact that the range of known concrete examples and construction techniques are limited. For general background on K¨ahler groups see [2] (and also [13, 4] for more recent results). Known constructions have shown that K¨ahler groups can present the following group theoretic properties: they can ● be non-residually finite [50] (see also [18]); ● be nilpotent of class 2 [14, 47]; ● admit a classifying space with finite k-skeleton, but no classifying space with finitely many k + 1-cells [23] (see also [5, 38, 12]); and arXiv:1807.03677v2 [math.GT] 7 Jun 2019 ● be non-coherent [33] (also [45, 27]). -
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. -
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 →∞. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups . -
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. -
PUSHING FILLINGS in RIGHT-ANGLED ARTIN GROUPS 11 a · E of XΓ
PUSHING FILLINGS IN RIGHT-ANGLED ARTIN GROUPS AARON ABRAMS, NOEL BRADY, PALLAVI DANI, MOON DUCHIN, AND ROBERT YOUNG Abstract. We construct pushing maps on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r2k+2. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties \at infinity," are defined here as a family of quasi-isometry invariants of groups. By pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V (2k+2)=k. We construct a class of RAAGs called orthoplex groups which show that each of these upper bounds is sharp. 1. Introduction Many of the groups studied in geometric group theory are subgroups of non-positively curved groups. This family includes lattices in symmetric spaces, Bestvina-Brady groups, and many solvable groups. In each of these cases, the group acts geometrically on a subset of a non-positively curved space, and one can approach the study of the subgroup by considering the geometry of that part of the model space. In this paper, we construct pushing maps for the cube complexes that model right-angled Artin groups. These maps serve to modify chains so that they lie in special subsets of the space. We find that the geometry of the groups is “flexible” enough that it is not much more difficult to fill curves and cycles in these special subsets than to fill them efficiently in the ambient space. -
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]. -
Arxiv:2012.00730V3 [Math.GR] 30 May 2021 Ab of the Dehn Function, Namely, the Abelianised Dehn Function Δh and the Homological Dehn Function FAH
HOMOLOGICAL DEHN FUNCTIONS OF GROUPS OF TYPE FP2 NOEL BRADY, ROBERT KROPHOLLER, AND IGNAT SOROKO Abstract. We prove foundational results for homological Dehn functions of groups of type FP2 such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions for Leary's groups GL(S) providing methods to obtain uncountably many groups with a given homological Dehn function. This allows us to show that there exist groups of type FP2 with quartic homological Dehn function and unsolvable word problem. 1. Introduction Geometric group theory can trace its roots back to the foundational work of Dehn [Deh11], in which he posed his famous three fundamental problems of group theory. The first of these problems is: The Word Problem. Let H be a finitely generated group with a generating set S. Is there an algorithm that takes as input words in S and outputs whether or not they are trivial? Dehn's seminal work showed that this problem is solvable for fundamental groups of sur- faces. However, in the 1950's Novikov [Nov55] and Boone [Boo58] showed that there exist finitely presented groups for which the word problem is unsolvable. For finitely presented groups the Dehn function δH determines exactly when such an algorithm exists. Moreover, there is an upper bound on the time complexity of such an algorithm in terms of δH . The Dehn function of finitely presented groups has received a great deal of interest over the last 100 years. For a more complete background we refer the reader to [Bri02] and the references therein.