The Topology, Geometry, and Dynamics of Free Groups
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Categories of Sets with a Group Action
Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. -
Groups Acting on Cat(0) Cube Complexes with Uniform Exponential Growth
GROUPS ACTING ON CAT(0) CUBE COMPLEXES WITH UNIFORM EXPONENTIAL GROWTH RADHIKA GUPTA, KASIA JANKIEWICZ, AND THOMAS NG Abstract. We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth. 1. Introduction In this article, we continue the inquiry to determine which groups that act on CAT(0) cube complexes have uniform exponential growth. Let G be a group with finite generating set S and corresponding Cayley graph Cay(G; S) equipped with the word metric. Let B(n; S) be the ball of radius n in Cay(G; S). The exponential growth rate of G with respect to S is defined as w(G; S) := lim jB(n; S)j1=n: n!1 The exponential growth rate of G is defined as w(G) := inf fw(G; S) j S finite generating setg : We say a group G has exponential growth if w(G; S) > 1 for some (hence every) finite generating set S. A group G is said to have uniform exponential growth if w(G) > 1. -
Lectures on Quasi-Isometric Rigidity Michael Kapovich 1 Lectures on Quasi-Isometric Rigidity 3 Introduction: What Is Geometric Group Theory? 3 Lecture 1
Contents Lectures on quasi-isometric rigidity Michael Kapovich 1 Lectures on quasi-isometric rigidity 3 Introduction: What is Geometric Group Theory? 3 Lecture 1. Groups and Spaces 5 1. Cayley graphs and other metric spaces 5 2. Quasi-isometries 6 3. Virtual isomorphisms and QI rigidity problem 9 4. Examples and non-examples of QI rigidity 10 Lecture 2. Ultralimits and Morse Lemma 13 1. Ultralimits of sequences in topological spaces. 13 2. Ultralimits of sequences of metric spaces 14 3. Ultralimits and CAT(0) metric spaces 14 4. Asymptotic Cones 15 5. Quasi-isometries and asymptotic cones 15 6. Morse Lemma 16 Lecture 3. Boundary extension and quasi-conformal maps 19 1. Boundary extension of QI maps of hyperbolic spaces 19 2. Quasi-actions 20 3. Conical limit points of quasi-actions 21 4. Quasiconformality of the boundary extension 21 Lecture 4. Quasiconformal groups and Tukia's rigidity theorem 27 1. Quasiconformal groups 27 2. Invariant measurable conformal structure for qc groups 28 3. Proof of Tukia's theorem 29 4. QI rigidity for surface groups 31 Lecture 5. Appendix 33 1. Appendix 1: Hyperbolic space 33 2. Appendix 2: Least volume ellipsoids 35 3. Appendix 3: Different measures of quasiconformality 35 Bibliography 37 i Lectures on quasi-isometric rigidity Michael Kapovich IAS/Park City Mathematics Series Volume XX, XXXX Lectures on quasi-isometric rigidity Michael Kapovich Introduction: What is Geometric Group Theory? Historically (in the 19th century), groups appeared as automorphism groups of some structures: • Polynomials (field extensions) | Galois groups. • Vector spaces, possibly equipped with a bilinear form| Matrix groups. -
Local Connectivity of Right-Angled Coxeter Group Boundaries
LOCAL CONNECTIVITY OF RIGHT-ANGLED COXETER GROUP BOUNDARIES MICHAEL MIHALIK, KIM RUANE, AND STEVE TSCHANTZ Abstract. We provide conditions on the defining graph of a right- angled Coxeter group presentation that guarantees the boundary of any CAT (0) space on which the group acts geometrically will be locally connected. 0. Introduction This is an edited version of the published version of our paper. The changes are minor, but clean up the paper quite a bit: the proof of lemma 5.8 is reworded and a figure is added showing where (in the published version) some undefined vertices of the Cayley graph are locate, the proof of theorem 3.2 is simplified by using lemma 4.2 (allowing us to eliminate four lemmas that appeared near the end of section 1). In [BM],the authors ask whether all one-ended word hyperbolic groups have locally connected boundary. In that paper, they relate the existence of global cut points in the boundary to local connectivity of the boundary. In [Bo], Bowditch gives a correspondence between local cut points in the bound- ary and splittings of the group over one-ended subgroups. In [S], Swarup uses the work of Bowditch and others to prove that the boundary of such a group cannot contain a global cut point, thus proving these boundaries are locally connected. The situation is quite different in the setting of CAT (0) groups. If G is a one-ended group acting geometrically on a CAT (0) space X, then @X can indeed be non-locally connected. For example, consider the group G = F2 × Z where F2 denotes the non-abelian free group of rank 2, acting on X = T ×R where T is the tree of valence 4 (or the Cayley graph of F2 with the standard generating set). -
Groups Acting on CAT (0) Cube Complexes with Uniform Exponential
GROUPS ACTING ON CAT(0) CUBE COMPLEXES WITH UNIFORM EXPONENTIAL GROWTH RADHIKA GUPTA, KASIA JANKIEWICZ, AND THOMAS NG Abstract. We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth. 1. Introduction In this article, we continue the inquiry to determine which groups that act on CAT(0) cube complexes have uniform exponential growth. Let G be a group with finite generating set S and corresponding Cayley graph Cay(G; S) equipped with the word metric. Let B(n; S) be the ball of radius n in Cay(G; S). The exponential growth rate of G with respect to S is defined as w(G; S) := lim jB(n; S)j1=n: n!1 The exponential growth rate of G is defined as w(G) := inf fw(G; S) j S finite generating setg : We say a group G has exponential growth if w(G; S) > 1 for some (hence every) finite generating set S. A group G is said to have uniform exponential growth if w(G) > 1. -
Homology of Hom Complexes
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 10 Homology of Hom Complexes Mychael Sanchez New Mexico State University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Sanchez, Mychael (2012) "Homology of Hom Complexes," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 10. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/10 Rose- Hulman Undergraduate Mathematics Journal Homology of Hom Complexes Mychael Sancheza Volume 13, No. 1, Spring 2012 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal New Mexico State University Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Homology of Hom Complexes Mychael Sanchez Abstract. The hom complex Hom (G; K) is the order complex of the poset com- posed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups. Acknowledgements: I acknowledge my advisor, Dr. Dan Ramras and the National Science Foundation for funding this research through Dr. Ramras' grants, DMS-1057557 and DMS-0968766. Page 212 RHIT Undergrad. Math. J., Vol. 13, No. 1 1 Introduction In 1978, László Lovász proved the Kneser Conjecture [8]. His proof involves associating to a graph G its neighborhood complex N (G) and using its topological properties to locate obstructions to graph colorings. This proof illustrates the fruitful interaction between graph theory, combinatorics and topology. -
Homology Homomorphism
|| Computations Induced (Co)homology homomorphism Daher Al Baydli Department of Mathematics National University of Ireland,Galway Feberuary, 2017 Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 1 / 11 Overview Resolution Chain and cochain complex homology cohomology Induce (Co)homology homomorphism Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 2 / 11 Resolution Definition Let G be a group and Z be the group of integers considered as a trivial ZG-module. The map : ZG −! Z from the integral group ring to Z, given by Σmg g 7−! Σmg , is called the augmentation. Definition Let G be a group. A free ZG-resolution of Z is an exact sequence of free G G @n+1 G @n G @n−1 @2 G @1 ZG-modules R∗ : · · · −! Rn+1 −! Rn −! Rn−1 −! · · · −! R1 −! G G R0 −! R−1 = Z −! 0 G with each Ri a free ZG-module for all i ≥ 0. Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 3 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. For each n 2 Z, the nth homology module of C is defined to be the quotient module Ker@n Hn(C) = Im@n+1 G We can also construct an induced cochain complex HomZG (R∗ ; A) of abelian groups: G G δn G δn−1 HomZG (R∗ ; A): · · · − HomZG (Rn+1; A) − HomZG (Rn ; A) − G δn−2 δ1 G δ0 G HomZG (Rn−1; A) − · · · − HomZG (R1 ; A) − HomZG (R0 ; A) Chain and cochain complex G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G @n+1 G @n abelian groups: R∗ ⊗ZG A : · · · −! Rn+1 ⊗ZG A −! Rn ⊗ZG A −! G @n−1 @2 G @1 G Rn−1 ⊗ZG A −! · · · −! R1 ⊗ZG A −! R0 ⊗ZG A Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. -
Homology Groups of Homeomorphic Topological Spaces
An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
Constructing Covers of the Rose
Constructing Covers of the rose. Robert Kropholler April 25, 2017 This was discussed in class but does not appear in any of the lecture notes. I will outline the constructions and proof here. Definition 0.1. Let F (X) be the free group on the set X. Assume jXj = n. Let H be any subgroup of F (X), we define the Schrier graph G(H) as follows: • The vertices of G(H) are the left cosets of H in G. • Two cosets gH; g0H are joined by an edge if there exists an x 2 X such that xgH = g0H. It should be noted that the Schrier graph of a normal subgroup N is the Cayley graph of G=N with respect to the generating set X. Let Rn be a cell complex with 1 vertex and n edges. We identify the fundamental group of Rn with F (X). Pick a bijection between the edges of X and give each edge an orientation. Proposition 0.2. The graph G(H) is a cover of Rn. Proof. We can label each edge with an element of X and give it an orientation so that the edge (gH; xgH) points towards xgH. We can then define a map p : G(H) ! Rn. This sends each vertex of G(H) to the unique vertex of Rn and each edge to the edge with same label by an orientation preserving homeomorphism. This defines a covering map. We must check that every point has a neighbourhood such that p is a homeomorphism on this neighbourhood. -
Uniform Exponential Growth of Non-Positively Curved Groups
Uniform exponential growth of non-positively curved groups A Dissertation Submitted to the Temple University Graduate Board in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY by Thomas Ng May, 2020 Dissertation Examining Committee: David Futer, Advisory Chair, Mathematics Matthew Stover, Mathematics Samuel Taylor, Mathematics Tarik Aougab, Haverford College, Mathematics and Statistics iii c by Thomas Ng May, 2020 All Rights Reserved iv ABSTRACT Uniform exponential growth of non-positively curved groups Thomas Ng DOCTOR OF PHILOSOPHY Temple University, May, 2020 Professor David Futer, Chair The ping-pong lemma was introduced by Klein in the late 1800s to show that certain subgroups of isometries of hyperbolic 3-space are free and remains one of very few tools that certify when a pair of group elements generate a free subgroup or semigroup. Quantitatively applying the ping-pong lemma to more general group actions on metric spaces requires a blend of understanding the large-scale global geometry of the underlying space with local combinatorial and dynamical behavior of the action. In the 1980s, Gromov publish a sequence of seminal works introducing several metric notions of non-positive curvature in group theory where he asked which finitely generated groups have uniform exponential growth. We give an overview of various developments of non- positive curvature in group theory and past results related to building free semigroups in the setting of non-positive curvature. We highlight joint work with Radhika Gupta and Kasia Jankiewicz and with Carolyn Abbott and v Davide Spriano that extends these tools and techniques to show several groups with that act on cube complexes and many hierarchically hyperbolic groups have uniform exponential growth. -
Moduli Spaces of Colored Graphs
MODULI SPACES OF COLORED GRAPHS MARKO BERGHOFF AND MAX MUHLBAUER¨ Abstract. We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by var- ious conditions on the coloring of their graphs. The motivation for this is to study Feynman integrals in quantum field theory using the combinatorial structure of these moduli spaces. Here a family of graphs is specified by the allowed Feynman diagrams in a particular quantum field theory such as (mas- sive) scalar fields or quantum electrodynamics. The resulting spaces are cell complexes with a rich and interesting combinatorial structure. We treat some examples in detail and discuss their topological properties, connectivity and homology groups. 1. Introduction The purpose of this article is to define and study moduli spaces of colored graphs in the spirit of [HV98] and [CHKV16] where such spaces for uncolored graphs were used to study the homology of automorphism groups of free groups. Our motivation stems from the connection between these constructions in geometric group theory and the study of Feynman integrals as pointed out in [BK15]. Let us begin with a brief sketch of these two fields and what is known so far about their relation. 1.1. Background. The analysis of Feynman integrals as complex functions of their external parameters (e.g. momenta and masses) is a special instance of a very gen- eral problem, understanding the analytic structure of functions defined by integrals. That is, given complex manifolds X; T and a function f : T ! C defined by Z f(t) = !t; Γ what can be deduced about f from studying the integration contour Γ ⊂ X and the integrand !t 2 Ω(X)? There is a mathematical treatment for a class of well-behaved cases [Pha11], arXiv:1809.09954v3 [math.AT] 2 Jul 2019 but unfortunately Feynman integrals are generally too complicated to answer this question thoroughly.