Algorithms, Dehn Functions, and Automatic 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 -
WHAT IS Outer Space?
WHAT IS Outer Space? Karen Vogtmann To investigate the properties of a group G, it is with a metric of constant negative curvature, and often useful to realize G as a group of symmetries g : S ! X is a homeomorphism, called the mark- of some geometric object. For example, the clas- ing, which is well-defined up to isotopy. From this sical modular group P SL(2; Z) can be thought of point of view, the mapping class group (which as a group of isometries of the upper half-plane can be identified with Out(π1(S))) acts on (X; g) f(x; y) 2 R2j y > 0g equipped with the hyper- by composing the marking with a homeomor- bolic metric ds2 = (dx2 + dy2)=y2. The study of phism of S { the hyperbolic metric on X does P SL(2; Z) and its subgroups via this action has not change. By deforming the metric on X, on occupied legions of mathematicians for well over the other hand, we obtain a neighborhood of the a century. point (X; g) in Teichm¨uller space. We are interested here in the (outer) automor- phism group of a finitely-generated free group. u v Although free groups are the simplest and most fundamental class of infinite groups, their auto- u v u v u u morphism groups are remarkably complex, and v x w v many natural questions about them remain unan- w w swered. We will describe a geometric object On known as Outer space , which was introduced in w w u v [2] to study Out(Fn). -
Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups
Automorphism groups of free groups, surface groups and free abelian groups Martin R. Bridson and Karen Vogtmann The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the map- ± ping class groups Mod (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a success- ful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similar- ities and differences between these series of groups and present some open problems motivated by this philosophy. ± Similarities among the groups Out(Fn), GL(n, Z) and Mod (Sg) begin with the fact that these are the outer automorphism groups of the most prim- itive types of torsion-free discrete groups, namely free groups, free abelian groups and the fundamental groups of closed orientable surfaces π1Sg. In the ± case of Out(Fn) and GL(n, Z) this is obvious, in the case of Mod (Sg) it is a classical theorem of Nielsen. -
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. -
THE SYMMETRIES of OUTER SPACE Martin R. Bridson* and Karen Vogtmann**
THE SYMMETRIES OF OUTER SPACE Martin R. Bridson* and Karen Vogtmann** ABSTRACT. For n 3, the natural map Out(Fn) → Aut(Kn) from the outer automorphism group of the free group of rank n to the group of simplicial auto- morphisms of the spine of outer space is an isomorphism. §1. Introduction If a eld F has no non-trivial automorphisms, then the fundamental theorem of projective geometry states that the group of incidence-preserving bijections of the projective space of dimension n over F is precisely PGL(n, F ). In the early nineteen seventies Jacques Tits proved a far-reaching generalization of this theorem: under suitable hypotheses, the full group of simplicial automorphisms of the spherical building associated to an algebraic group is equal to the algebraic group — see [T, p.VIII]. Tits’s theorem implies strong rigidity results for lattices in higher-rank — see [M]. There is a well-developed analogy between arithmetic groups on the one hand and mapping class groups and (outer) automorphism groups of free groups on the other. In this analogy, the role played by the symmetric space in the classical setting is played by the Teichmuller space in the case of case of mapping class groups and by Culler and Vogtmann’s outer space in the case of Out(Fn). Royden’s Theorem (see [R] and [EK]) states that the full isometry group of the Teichmuller space associated to a compact surface of genus at least two (with the Teichmuller metric) is the mapping class group of the surface. An elegant proof of Royden’s theorem was given recently by N. -
The Cohomology of Automorphism Groups of Free Groups
The cohomology of automorphism groups of free groups Karen Vogtmann∗ Abstract. There are intriguing analogies between automorphism groups of finitely gen- erated free groups and mapping class groups of surfaces on the one hand, and arithmetic groups such as GL(n, Z) on the other. We explore aspects of these analogies, focusing on cohomological properties. Each cohomological feature is studied with the aid of topolog- ical and geometric constructions closely related to the groups. These constructions often reveal unexpected connections with other areas of mathematics. Mathematics Subject Classification (2000). Primary 20F65; Secondary, 20F28. Keywords. Automorphism groups of free groups, Outer space, group cohomology. 1. Introduction In the 1920s and 30s Jakob Nielsen, J. H. C. Whitehead and Wilhelm Magnus in- vented many beautiful combinatorial and topological techniques in their efforts to understand groups of automorphisms of finitely-generated free groups, a tradition which was supplemented by new ideas of J. Stallings in the 1970s and early 1980s. Over the last 20 years mathematicians have been combining these ideas with others motivated by both the theory of arithmetic groups and that of surface mapping class groups. The result has been a surge of activity which has greatly expanded our understanding of these groups and of their relation to many areas of mathe- matics, from number theory to homotopy theory, Lie algebras to bio-mathematics, mathematical physics to low-dimensional topology and geometric group theory. In this article I will focus on progress which has been made in determining cohomological properties of automorphism groups of free groups, and try to in- dicate how this work is connected to some of the areas mentioned above. -
Perspectives in Lie Theory Program of Activities
INdAM Intensive research period Perspectives in Lie Theory Program of activities Session 3: Algebraic topology, geometric and combinatorial group theory Period: February 8 { February 28, 2015 All talks will be held in Aula Dini, Palazzo del Castelletto. Monday, February 9, 2015 • 10:00- 10:40, registration • 10:40, coffee break. • 11:10- 12:50, Vic Reiner, Reflection groups and finite general linear groups, lecture 1. • 15:00- 16:00, Michael Falk, Rigidity of arrangement complements • 16:00, coffee break. • 16:30- 17:30, Max Wakefield, Kazhdan-Lusztig polynomial of a matroid • 17:30- 18:30, Angela Carnevale, Odd length: proof of two conjectures and properties (young semi- nar). • 18:45- 20:30, Welcome drink (Sala del Gran Priore, Palazzo della Carovana, Piazza dei Cavalieri) Tuesday, February 10, 2015 • 9:50-10:40, Ulrike Tillmann, Homology of mapping class groups and diffeomorphism groups, lecture 1. • 10:40, coffee break. • 11:10- 12:50, Karen Vogtmann, On the cohomology of automorphism groups of free groups, lecture 1. • 15:00- 16:00, Tony Bahri,New approaches to the cohomology of polyhedral products • 16:00, coffee break. • 16:30- 17:30, Alexandru Dimca, On the fundamental group of algebraic varieties • 17:30- 18:30, Nancy Abdallah, Cohomology of Algebraic Plane Curves (young seminar). Wednesday, February 11, 2015 • 9:00- 10:40, Vic Reiner, Reflection groups and finite general linear groups, lecture 2. • 10:40, coffee break. • 11:10- 12:50, Ulrike Tillmann, Homology of mapping class groups and diffeomorphism groups, lec- ture 2 • 15:00- 16:00, Karola Meszaros, Realizing subword complexes via triangulations of root polytopes • 16:00, coffee break. -
Karen Vogtmann
CURRICULUM VITAE -KAREN VOGTMANN Mathematics Institute Office: C2.05 Zeeman Bldg. Phone: +44 (0) 2476 532739 University of Warwick Email: [email protected] Coventry CV4 7AL PRINCIPAL FIELDS OF INTEREST Geometric group theory, Low-dimensional topology, Cohomology of groups EDUCATION B.A. University of California, Berkeley 1971 Ph.D. University of California, Berkeley 1977 ACADEMIC POSITIONS University of Warwick, Professor, 9/13 to present Cornell University – Goldwin Smith Professor of Mathematics Emeritus, 7/15 to present – Goldwin Smith Professor of Mathematics, 7/11 to 7/15 – Professor, 1/94 to 7/11 – Associate Professor, 7/87 to 12/93 – Assistant Professor, 7/85 to 6/87 – Visiting Assistant Professor, 9/84 to 6/85 Columbia University, Assistant Professor, 7/79 to 6/86 Brandeis University, Visiting Assistant Professor, 9/78 to 12/78 University of Michigan, Visiting Assistant Professor, 9/77 to 6/78 and 1/79 to 6/79 RESEARCH AND SABBATICAL POSITIONS MSRI, Berkeley, CA 8/19 to 11/19 Newton Institute, Cambridge, Mass, 3/17 to 5/17 MSRI, Berkeley, CA 8/16 to 12/16 Research Professor, ICERM, Providence, RI, 9/13 to 12/13 Freie Universitat¨ Berlin, Berlin, Germany, 6/12 Mittag-Leffler Institute, Stockholm, Sweden, 3/12 to 5/12 Visiting Researcher, Oxford University, Oxford, England, 2/12 Professeur invite, Marseilles, France, 5/11 Hausdorff Institute for Mathematics, 9/09 to 12/09 and 5/10-8/10 Mathematical Sciences Research Institute, Berkeley, CA, 8/07-12/07 I.H.E.S., Bures-sur-Yvette, France 3/04 Professeur Invite,´ Marseilles, France, 3/00 Mathematical Sciences Research Institute, Berkeley, 1/95 to 7/95 I.H.E.S., Bures-sur-Yvette, France, 1/93-8/93 Chercheur, C.N.R.S., E.N.S. -
2020 Leroy P. Steele Prizes
FROM THE AMS SECRETARY 2020 Leroy P. Steele Prizes The 2020 Leroy P. Steele Prizes were presented at the 126th Annual Meeting of the AMS in Denver, Colorado, in Jan- uary 2020. The Steele Prize for Mathematical Exposition was awarded to Martin R. Bridson and André Haefliger; the Prize for Seminal Contribution to Research in Analysis/Probability was awarded to Craig Tracy and Harold Widom; and the Prize for Lifetime Achievement was awarded to Karen Uhlenbeck. Citation for Riemannian geometry and group theory, that the field of Mathematical Exposition: geometric group theory came into being. Much of the 1990s Martin R. Bridson was spent finding rigorous proofs of Gromov’s insights and André Haefliger and expanding upon them. Metric Spaces of Non-Positive The 2020 Steele Prize for Math- Curvature is the outcome of that decade of work, and has ematical Exposition is awarded been the standard textbook and reference work throughout to Martin R. Bridson and André the field in the two decades of dramatic progress since its Haefliger for the book Metric publication in 1999. Spaces of Non-Positive Curvature, A metric space of non-positive curvature is a geodesic published by Springer-Verlag metric space satisfying (local) CAT(0) condition, that every in 1999. pair of points on a geodesic triangle should be no further Metric Spaces of Non-Positive apart than the corresponding points on the “comparison Martin R. Bridson Curvature is the authoritative triangle” in the Euclidean plane. Examples of such spaces reference for a huge swath of include non-positively curved Riemannian manifolds, modern geometric group the- Bruhat–Tits buildings, and a wide range of polyhedral ory. -
Hyperbolic Groups and Completely Simple Semigroups
March 25, 2004 11:52 WSPC/Trim Size: 9in x 6in for Proceedings hypcssgps3 HYPERBOLIC GROUPS AND COMPLETELY SIMPLE SEMIGROUPS John Fountain Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. e-mail : [email protected] Mark Kambites School of Mathematics & Statistics, Carleton University, Herzberg Laboratories, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada e-mail : [email protected] We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Sch¨utzenberger graphs; having a context-free multipli- cation table; or having word hyperbolic maximal subgroups. Our main result is that these conditions coincide in the case of finitely generated completely simple semigroups. This paper is based on a lecture given at the workshop on Semigroups and Languages held at the Centro de Algebra´ da Universidade de Lisboa in November 2002. The aim of the paper, as of the talk, is to provide semigroup theorists with a gentle introduction to the concept of a word hyperbolic group (hereafter referred to as simply a ‘hyperbolic group’), to discuss recent work of Gilman which makes it possible to introduce a notion of (word) hyperbolic semigroup and then to examine these ideas in the context of completely simple semigroups. The first three sections of the paper are expository while the fourth introduces new results. After briefly discussing Dehn’s algorithm, we describe hyperbolic groups in terms of ‘slim triangles’, and then describe some properties of these groups which we need later in the paper. -
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.