Covering Group Theory for Compact Groups
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An Overview of Topological Groups: Yesterday, Today, Tomorrow
axioms Editorial An Overview of Topological Groups: Yesterday, Today, Tomorrow Sidney A. Morris 1,2 1 Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia; [email protected]; Tel.: +61-41-7771178 2 Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia Academic Editor: Humberto Bustince Received: 18 April 2016; Accepted: 20 April 2016; Published: 5 May 2016 It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. -
the Group of Homeomorphisms of a Solenoid Which Are Isotopic to The
. The Group of Homeomorphisms of a Solenoid which are Isotopic to the Identity A DISSERTATION Submitted to the Centro de Investigaci´on en Matem´aticas in partial fulfillment of the requirements for the degree of DOCTOR IN SCIENCES Field of Mathematics by Ferm´ınOmar Reveles Gurrola Advisors Dr. Manuel Cruz L´opez Dr. Xavier G´omez–Mont Avalos´ GUANAJUATO, GUANAJUATO September 2015 2 3 A mi Maestro Cr´uz–L´opez que me ensen´oTodo˜ lo que S´e; a mi Maestra, que me Ensenar´atodo˜ lo que No s´e; a Ti Maestro Cantor, mi Vida, mi Muerte Chiquita. 4 Abstract In this work, we present a detailed study of the connected component of the identity of the group of homeomorphisms of a solenoid Homeo+(S). We treat the universal one–dimensional solenoid S as the universal algebraic covering of the circle S1; that is, as the inverse limit of all the n–fold coverings of S1. Moreover, this solenoid is a foliated space whose leaves are homeomorphic to R and a typical transversal is isomorphic to the completion of the integers Zb. We are mainly interested in the homotopy type of Homeo+(S). Using the theory of cohomology group we calculate its second cohomology groups with integer and real coefficients. In fact, we are able to calculate the associated bounded cohomology groups. That is, we found the Euler class for the universal central extension of Homeo+(S), which is constructed via liftings to the covering space R × Zb of S. We show that this is a bounded cohomology class. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
SPACES with a COMPACT LIE GROUP of TRANSFORMATIONS A
SPACES WITH A COMPACT LIE GROUP OF TRANSFORMATIONS a. m. gleason Introduction. A topological group © is said to act on a topological space 1{ if the elements of © are homeomorphisms of 'R. onto itself and if the mapping (<r, p)^>a{p) of ©X^ onto %_ is continuous. Familiar examples include the rotation group acting on the Car- tesian plane and the Euclidean group acting on Euclidean space. The set ®(p) (that is, the set of all <r(p) where <r(E®) is called the orbit of p. If p and q are two points of fx, then ®(J>) and ©(g) are either identical or disjoint, hence 'R. is partitioned by the orbits. The topological structure of the partition becomes an interesting question. In the case of the rotations of the Cartesian plane we find that, excising the singularity at the origin, the remainder of space is fibered as a direct product. A similar result is easily established for a compact Lie group acting analytically on an analytic manifold. In this paper we make use of Haar measure to extend this result to the case of a compact Lie group acting on any completely regular space. The exact theorem is given in §3. §§1 and 2 are preliminaries, while in §4 we apply the main result to the study of the structure of topo- logical groups. These applications form the principal motivation of the entire study,1 and the author hopes to develop them in greater detail in a subsequent paper. 1. An extension theorem. In this section we shall prove an exten- sion theorem which is an elementary generalization of the well known theorem of Tietze. -
The Geometry and Fundamental Groups of Solenoid Complements
November 5, 2018 THE GEOMETRY AND FUNDAMENTAL GROUPS OF SOLENOID COMPLEMENTS GREGORY R. CONNER, MARK MEILSTRUP, AND DUSANˇ REPOVSˇ Abstract. A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements. In this paper we study 3-manifolds which are complements of solenoids in S3. This theory is a natural extension of the study of knot complements in S3; many of the tools that we use are the same as those used in knot theory and braid theory. We will mainly be concerned with studying the geometry and fundamental groups of 3-manifolds which are solenoid complements. We review basic information about solenoids in section 1. In section 2 we discuss the calculation of the fundamental group of solenoid complements. In section 3 we show that every solenoid has an embedding in S3 so that the complementary 3-manifold has an Abelian fundamental group, which is in fact a subgroup of Q (Theorem 3.5). In section 4 we show that each solenoid has an embedding whose complement has a non-Abelian fundamental group (Theorem 4.3). -
Horocycle Flows for Laminations by Hyperbolic Riemann Surfaces and Hedlund’S Theorem
JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2016.10.113 VOLUME 10, 2016, 113–134 HOROCYCLE FLOWS FOR LAMINATIONS BY HYPERBOLIC RIEMANN SURFACES AND HEDLUND’S THEOREM MATILDE MARTíNEZ, SHIGENORI MATSUMOTO AND ALBERTO VERJOVSKY (Communicated by Federico Rodriguez Hertz) To Étienne Ghys, on the occasion of his 60th birthday ABSTRACT. We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle (Mˆ ,T 1F ) of a compact minimal lamination (M,F ) by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if F has a leaf which is not simply connected, the horocyle flow is topologically transitive. 1. INTRODUCTION The geodesic and horocycle flows over compact hyperbolic surfaces have been studied in great detail since the pioneering work in the 1930’s by E. Hopf and G. Hedlund. Such flows are particular instances of flows on homogeneous spaces induced by one-parameter subgroups, namely, if G is a Lie group, K a closed subgroup and N a one-parameter subgroup of G, then N acts on the homogeneous space K \G by right multiplication on left cosets. One very important case is when G SL(n,R), K SL(n,Z) and N is a unipo- Æ Æ tent one parameter subgroup of SL(n,R), i.e., all elements of N consist of ma- trices having all eigenvalues equal to one. In this case SL(n,Z)\SL(n,R) is the space of unimodular lattices. -
3-Manifolds That Admit Knotted Solenoids As Attractors
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 356, Number 11, Pages 4371{4382 S 0002-9947(04)03503-2 Article electronically published on February 27, 2004 3-MANIFOLDS THAT ADMIT KNOTTED SOLENOIDS AS ATTRACTORS BOJU JIANG, YI NI, AND SHICHENG WANG Abstract. Motivated by the study in Morse theory and Smale's work in dy- namics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non- wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds. 1. Introduction The solenoids were first defined in mathematics by Vietoris in 1927 for 2-adic case and by others later in general case, which can be presented either in an abstract way (inverse limit of self-coverings of circles) or in a geometric way (nested intersections of solid tori). The solenoids were introduced into dynamics by Smale as hyperbolic attractors in his celebrated paper [S]. Standard notions in dynamics and in 3-manifold topology will be given in Section 2. The new definitions are the following: Let N = S1 × D2,whereS1 is the unit circle and D2 is the unit disc. Both S1 and D2 admit \linear structures". Let e : N ! N be a \linear", D2-level-preserving embedding such that (a) e(S1 ×∗)isaw-string braid in N for each ∗∈D2,where w>1 in an integer; (b) for each θ 2 S1, the radius of e(θ × D2)is1=w2. -
Solvable Groups Whose Character Degree Graphs Generalize Squares
J. Group Theory 23 (2020), 217–234 DOI 10.1515/jgth-2019-0027 © de Gruyter 2020 Solvable groups whose character degree graphs generalize squares Mark L. Lewis and Qingyun Meng Communicated by Robert M. Guralnick Abstract. Let G be a solvable group, and let .G/ be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that .G/ is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and .G/ is block square, then G is a direct product of subgroups having disconnected character degree graphs. 1 Introduction Throughout this paper, G will be a finite group, and cd.G/ .1/ Irr.G/ D ¹ j 2 º will be the set of irreducible character degrees of G. The idea of studying the in- fluence of cd.G/ on the structure of G via graphs related to cd.G/ was introduced in [9]. The prime divisor character degree graph which is denoted by .G/ for G. We will often shorten the name of this graph to the character degree graph. The vertex set for .G/ is .G/, the set of all the prime divisors of cd.G/, and there is an edge between two vertices p and q if pq divides some degree a cd.G/. 2 The character degree graph has proven to be a useful tool to study the normal structure of finite solvable groups when given information about cd.G/. -
Finite Groups with Unbalancing 2-Components of {L(4), He]-Type
JOKRXAL OF ALGEBRA 60, 96-125 (1979) Finite Groups with Unbalancing 2-Components of {L(4), He]-Type ROBERT L. GKIE~S, JR.* AND RONALI) SOLOMON” Department of Mathematics, linivusity of BZichigan, Ann Arbor, Mizhigan 48109 rind Department of Mathematics, Ohio State IJniver.&y, Columbw, Ohio 43210 Communicated by Walter Feit Received September 18, 1977 1. INTR~DUCTJ~~ Considerable progress has been made in recent years towards the classification of finite simple grnups of odd Chevallcy type These groups may be defined as simple groups G containing an involution t such that C,(t)jO(C,(t)) has a subnormal quasi-simple subgroup. This property is possessed by most Chevalley groups over fields of odd order but by no simple Chevalley groups over fields of characteristic 2, hence the name. It is also a property of alternating groups of degree at least 9 and eighteen of the known sporadic simple groups. Particular attention in this area has focussed on the so-called B-Conjecture. R-Conjecture: Let G be a finite group with O(G) = (1 j. Let t be an involu- tion of G andL a pcrfcct subnormal subgroup of CJt) withL/O(fJ) quasi-simple, then L is quasi-simple. The R-Conjecture would follow as an easy corollary of the Unbalanced Group Conjecture (.5’-Conjecture). U-Chjecture: Let G be a finite group withF*(G) quasi-simple. Suppose there is an involution t of G with O(C,(t)) e O(G). Then F*(G)/F(G) is isomorphic to one of the following: (I) A Chevalley group of odd characteristic. -
Representation Theory of Compact Groups - Lecture Notes up to 08 Mar 2017
Representation theory of compact groups - Lecture notes up to 08 Mar 2017 Alexander Yom Din March 9, 2017 These notes are not polished yet; They are not very organized, and might contain mistakes. 1 Introduction 1 ∼ ∼ × Consider the circle group S = R=Z = C1 - it is a compact group. One studies the space L2(S1) of square-integrable complex-valued fucntions on the circle. We have an action of S1 on L2(S1) given by (gf)(x) = f(g−1x): We can suggestively write ^ Z 2 1 L (S ) = C · δx; x2S1 where gδx = δgx: Thus, the "basis" of delta functions is "permutation", or "geometric". Fourier theory describes another, "spectral" basis. Namely, we consider the functions 2πinx χn : x (mod 1) 7! e where n 2 Z. The main claim is that these functions form a Hilbert basis for L2(S1), hence we can write ^ 2 1 M L (S ) = C · χn; n2Z where −1 gχn = χn (g)χn: In other words, we found a basis of eigenfunctions for the translation operators. 1 What we would like to do in the course is to generalize the above picture to more general compact groups. For a compact group G, one considers L2(G) with the action of G × G by translation on the left and on the right: −1 ((g1; g2)f)(x) = f(g1 xg2): When the group is no longer abelian, we will not be able to find enough eigen- functions for the translation operators to form a Hilbert basis. Eigenfunctions can be considered as spanning one-dimensional invariant subspaces, and what we will be able to find is "enough" finite-dimensional "irreducible" invariant sub- spaces (which can not be decomposed into smaller invariant subspaces). -
On the Dynamics of G-Solenoids. Applications to Delone Sets
Ergod. Th. & Dynam. Sys. (2003), 23, 673–691 c 2003 Cambridge University Press DOI: 10.1017/S0143385702001578 Printed in the United Kingdom On the dynamics of G-solenoids. Applications to Delone sets RICCARDO BENEDETTI† and JEAN-MARC GAMBAUDO‡ † Dipartimento di Matematica, Universita` di Pisa, via F. Buonarroti 2, 56127 Pisa, Italy (e-mail: [email protected]) ‡ Institut de Mathematiques´ de Bourgogne, U.M.R. 5584 du CNRS, Universite´ de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France (e-mail: [email protected]) (Received 2 September 2002 and accepted in revised form 22 January 2003) Abstract.AG-solenoid is a laminated space whose leaves are copies of a single Lie group G and whose transversals are totally disconnected sets. It inherits a G-action and can be considered as a dynamical system. Free Zd -actions on the Cantor set as well as a large class of tiling spaces possess such a structure of G-solenoids. For a large class of Lie groups, we show that a G-solenoid can be seen as a projective limit of branched manifolds modeled on G. This allows us to give a topological description of the transverse invariant measures associated with a G-solenoid in terms of a positive cone in the projective limit of the dim(G)-homology groups of these branched manifolds. In particular, we exhibit a simple criterion implying unique ergodicity. Particular attention is paid to the case when the Lie group G is the group of affine orientation-preserving isometries of the Euclidean space or its subgroup of translations. 1. -
COMPACT LIE GROUPS Contents 1. Smooth Manifolds and Maps 1 2
COMPACT LIE GROUPS NICHOLAS ROUSE Abstract. The first half of the paper presents the basic definitions and results necessary for investigating Lie groups. The primary examples come from the matrix groups. The second half deals with representation theory of groups, particularly compact groups. The end result is the Peter-Weyl theorem. Contents 1. Smooth Manifolds and Maps 1 2. Tangents, Differentials, and Submersions 3 3. Lie Groups 5 4. Group Representations 8 5. Haar Measure and Applications 9 6. The Peter-Weyl Theorem and Consequences 10 Acknowledgments 14 References 14 1. Smooth Manifolds and Maps Manifolds are spaces that, in a certain technical sense defined below, look like Euclidean space. Definition 1.1. A topological manifold of dimension n is a topological space X satisfying: (1) X is second-countable. That is, for the space's topology T , there exists a countable base, which is a countable collection of open sets fBαg in X such that every set in T is the union of a subcollection of fBαg. (2) X is a Hausdorff space. That is, for every pair of points x; y 2 X, there exist open subsets U; V ⊆ X such that x 2 U, y 2 V , and U \ V = ?. (3) X is locally homeomorphic to open subsets of Rn. That is, for every point x 2 X, there exists a neighborhood U ⊆ X of x such that there exists a continuous bijection with a continuous inverse (i.e. a homeomorphism) from U to an open subset of Rn. The first two conditions preclude pathological spaces that happen to have a homeomorphism into Euclidean space, and force a topological manifold to share more properties with Euclidean space.