ISOMETRY GROUPS of COMPACT RIEMANN SURFACES Contents 1
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1.6 Covering Spaces 1300Y Geometry and Topology
1.6 Covering spaces 1300Y Geometry and Topology 1.6 Covering spaces Consider the fundamental group π1(X; x0) of a pointed space. It is natural to expect that the group theory of π1(X; x0) might be understood geometrically. For example, subgroups may correspond to images of induced maps ι∗π1(Y; y0) −! π1(X; x0) from continuous maps of pointed spaces (Y; y0) −! (X; x0). For this induced map to be an injection we would need to be able to lift homotopies in X to homotopies in Y . Rather than consider a huge category of possible spaces mapping to X, we restrict ourselves to a category of covering spaces, and we show that under some mild conditions on X, this category completely encodes the group theory of the fundamental group. Definition 9. A covering map of topological spaces p : X~ −! X is a continuous map such that there S −1 exists an open cover X = α Uα such that p (Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα. We then refer to (X;~ p) (or simply X~, abusing notation) as a covering space of X. Let (X~i; pi); i = 1; 2 be covering spaces of X. A morphism of covering spaces is a covering map φ : X~1 −! X~2 such that the diagram commutes: φ X~1 / X~2 AA } AA }} p1 AA }}p2 A ~}} X We will be considering covering maps of pointed spaces p :(X;~ x~0) −! (X; x0), and pointed morphisms between them, which are defined in the obvious fashion. -
On Isometries on Some Banach Spaces: Part I
Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections On isometries on some Banach spaces { something old, something new, something borrowed, something blue, Part I Dijana Iliˇsevi´c University of Zagreb, Croatia Recent Trends in Operator Theory and Applications Memphis, TN, USA, May 3{5, 2018 Recent work of D.I. has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Something old, something new, something borrowed, something blue is referred to the collection of items that helps to guarantee fertility and prosperity. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. A linear map ': X!Y is called a linear isometry if k'(x)k = jxj; x 2 X : We shall be interested in surjective linear isometries on Banach spaces. One of the main problems is to give explicit description of isometries on a particular space. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. -
Computing the Complexity of the Relation of Isometry Between Separable Banach Spaces
Computing the complexity of the relation of isometry between separable Banach spaces Julien Melleray Abstract We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [1] and Mayer-Wolf [4]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. 1 Introduction Over the past fteen years or so, the theory of complexity of Borel equivalence relations has been a very active eld of research; in this paper, we compute the complexity of a relation of geometric nature, the relation of (linear) isom- etry between separable Banach spaces. Before stating precisely our result, we begin by quickly recalling the basic facts and denitions that we need in the following of the article; we refer the reader to [3] for a thorough introduction to the concepts and methods of descriptive set theory. Before going on with the proof and denitions, it is worth pointing out that this article mostly consists in putting together various results which were already known, and deduce from them after an easy manipulation the complexity of the relation of isometry between separable Banach spaces. Since this problem has been open for a rather long time, it still seems worth it to explain how this can be done, and give pointers to the literature. Still, it seems to the author that this will be mostly of interest to people with a knowledge of descriptive set theory, so below we don't recall descriptive set-theoretic notions but explain quickly what Lipschitz-free Banach spaces are and how that theory works. -
[Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group. -
Isometries and the Plane
Chapter 1 Isometries of the Plane \For geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. (W. K. Clifford) The focus of this first chapter is the 2-dimensional real plane R2, in which a point P can be described by its coordinates: 2 P 2 R ;P = (x; y); x 2 R; y 2 R: Alternatively, we can describe P as a complex number by writing P = (x; y) = x + iy 2 C: 2 The plane R comes with a usual distance. If P1 = (x1; y1);P2 = (x2; y2) 2 R2 are two points in the plane, then p 2 2 d(P1;P2) = (x2 − x1) + (y2 − y1) : Note that this is consistent withp the complex notation. For P = x + iy 2 C, p 2 2 recall that jP j = x + y = P P , thus for two complex points P1 = x1 + iy1;P2 = x2 + iy2 2 C, we have q d(P1;P2) = jP2 − P1j = (P2 − P1)(P2 − P1) p 2 2 = j(x2 − x1) + i(y2 − y1)j = (x2 − x1) + (y2 − y1) ; where ( ) denotes the complex conjugation, i.e. x + iy = x − iy. We are now interested in planar transformations (that is, maps from R2 to R2) that preserve distances. 1 2 CHAPTER 1. ISOMETRIES OF THE PLANE Points in the Plane • A point P in the plane is a pair of real numbers P=(x,y). d(0,P)2 = x2+y2. • A point P=(x,y) in the plane can be seen as a complex number x+iy. -
Isometry and Automorphisms of Constant Dimension Codes
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Constant Dimension Codes Anna-Lena Trautmann Institute of Mathematics University of Zurich \Crypto and Coding" Z¨urich, March 12th 2012 1 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 2 / 25 isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding 3 / 25 automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes 3 / 25 automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding 3 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes 3 / 25 Definition Subspace metric: dS(U; V) = dim(U + V) − dim(U\V) Injection metric: dI (U; V) = max(dim U; dim V) − dim(U\V) Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes Definition n n The projective geometry P(Fq ) is the set of all subspaces of Fq . -
Sufficient Conditions for Periodicity of a Killing Vector Field Walter C
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 3, May 1973 SUFFICIENT CONDITIONS FOR PERIODICITY OF A KILLING VECTOR FIELD WALTER C. LYNGE Abstract. Let X be a complete Killing vector field on an n- dimensional connected Riemannian manifold. Our main purpose is to show that if X has as few as n closed orbits which are located properly with respect to each other, then X must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold M. We give a generalization of this characterization which applies to arbitrary complete vector fields on M. Theorem. Let X be a complete Killing vector field on a connected, n- dimensional Riemannian manifold M. Assume there are n distinct points p, Pit' " >Pn-i m M such that the respective orbits y, ylt • • • , yn_x of X through them are closed and y is nontrivial. Suppose further that each pt is joined to p by a unique minimizing geodesic and d(p,p¡) = r¡i<.D¡2, where d denotes distance on M and D is the diameter of y as a subset of M. Let_wx, • ■ ■ , wn_j be the unit vectors in TVM such that exp r¡iwi=pi, i=l, • ■ • , n— 1. Assume that the vectors Xv, wx, ■• • , wn_x span T^M. Then the flow cptof X is periodic. Proof. Fix /' in {1, 2, •••,«— 1}. We first show that the orbit yt does not lie entirely in the sphere Sj,(r¡¿) of radius r¡i about p. -
Lecture 34: Hadamaard's Theorem and Covering Spaces
Lecture 34: Hadamaard’s Theorem and Covering spaces In the next few lectures, we’ll prove the following statement: Theorem 34.1. Let (M,g) be a complete Riemannian manifold. Assume that for every p M and for every x, y T M, we have ∈ ∈ p K(x, y) 0. ≤ Then for any p M,theexponentialmap ∈ exp : T M M p p → is a local diffeomorphism, and a covering map. 1. Complete Riemannian manifolds Recall that a connected Riemannian manifold (M,g)iscomplete if the geodesic equation has a solution for all time t.Thisprecludesexamplessuch as R2 0 with the standard Euclidean metric. −{ } 2. Covering spaces This, not everybody is familiar with, so let me give a quick introduction to covering spaces. Definition 34.2. Let p : M˜ M be a continuous map between topological spaces. p is said to be a covering→ map if (1) p is a local homeomorphism, and (2) p is evenly covered—that is, for every x M,thereexistsaneighbor- 1 ∈ hood U M so that p− (U)isadisjointunionofspacesU˜ for which ⊂ α p ˜ is a homeomorphism onto U. |Uα Example 34.3. Here are some standard examples: (1) The identity map M M is a covering map. (2) The obvious map M →... M M is a covering map. → (3) The exponential map t exp(it)fromR to S1 is a covering map. First, it is clearly a local → homeomorphism—a small enough open in- terval in R is mapped homeomorphically onto an open interval in S1. 113 As for the evenly covered property—if you choose a proper open in- terval inside S1, then its preimage under the exponential map is a disjoint union of open intervals in R,allhomeomorphictotheopen interval in S1 via the exponential map. -
Isometry Test for Coordinate Transformations Sudesh Kumar Arora
Isometry test for coordinate transformations Sudesh Kumar Arora Isometry test for coordinate transformations Sudesh Kumar Arora∗y Date January 30, 2020 Abstract A simple scheme is developed to test whether a given coordinate transformation of the metric tensor is an isometry or not. The test can also be used to predict the time dilation caused by any type of motion for a given space-time metric. Keywords and phrases: general relativity; coordinate transformation; diffeomorphism; isometry; symmetries. 1 Introduction Isometry is any coordinate transformation which preserves the distances. In special relativity, Minkowski metric has Poincare as the group of isometries, and the corresponding coordinate transformations leave the physical properties of the metric like distance invariant. In general relativity any infinitesimal trans- formation along Killing vector fields is an isometry. These are infinitesimal transformations generated by one parameter x0µ = xµ + V µ (1) Here V (x) = V µ@µ can be thought of like a vector field with components given as dx0µ V µ(x) = (x) (2) d All coordinate transformations generated such can be checked for isometry by calculating Lie deriva- tive of the metric tensors along the vector field V . If the Lie derivative vanishes then the transformation is an isometry (1; 2) and vector field V is called Killing Vector field. Killing vector fields are used to study the symmetries of the space-time metrics and the associated conserved quantities. But there are some challenges in using Killing equations for checking isometry. Coordinate transformations which are either discrete (not generated by the one-parameter ) or not possible to define as integral curves of a vectors field cannot be checked for isometry using this scheme. -
COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U. -
Generalized Covering Spaces and the Galois Fundamental Group
GENERALIZED COVERING SPACES AND THE GALOIS FUNDAMENTAL GROUP CHRISTIAN KLEVDAL Contents 1. Introduction 1 2. A Category of Covers 2 3. Uniform Spaces and Topological Groups 6 4. Galois Theory of Semicovers 9 5. Functoriality of the Galois Fundamental Group 15 6. Universal Covers 16 7. The Topologized Fundamental Group 17 8. Covers of the Earring 21 9. The Galois Fundamental Group of the Harmonic Archipelago 22 10. The Galois Fundamental Group of the Earring 23 References 24 Gal Abstract. We introduce a functor π1 from the category of based, connected, locally path connected spaces to the category of complete topological groups. We then compare this groups to the fundamental group. In particular, we show that there is a topological σ Gal group π1 (X; x) whose underlying group is π1(X; x) so that π1 (X; x) is the completion σ of π1 (X; x). 1. Introduction Gal In this paper, we give define a topological group π1 (X; x) called the Galois funda- mental group, which is associated to any based, connected, locally path connected space X. As the notation suggests, this group is intimately related to the fundamental group π1(X; x). The motivation of the Galois fundamental group is to look at automorphisms of certain covers of a space instead of homotopy classes of loops in the space. It is well known that if X has a universal cover, then π1(X; x) is isomorphic to the group of deck transformations of the universal cover. However, this approach breaks down if X is not semi locally simply connected. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups.