Compact Groups and Fixed Point Sets
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Faithful Abelian Groups of Infinite Rank Ulrich Albrecht
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 1, May 1988 FAITHFUL ABELIAN GROUPS OF INFINITE RANK ULRICH ALBRECHT (Communicated by Bhama Srinivasan) ABSTRACT. Let B be a subgroup of an abelian group G such that G/B is isomorphic to a direct sum of copies of an abelian group A. For B to be a direct summand of G, it is necessary that G be generated by B and all homomorphic images of A in G. However, if the functor Hom(A, —) preserves direct sums of copies of A, then this condition is sufficient too if and only if M ®e(A) A is nonzero for all nonzero right ¿ï(A)-modules M. Several examples and related results are given. 1. Introduction. There are only very few criteria for the splitting of exact sequences of torsion-free abelian groups. The most widely used of these was given by Baer in 1937 [F, Proposition 86.5]: If G is a pure subgroup of a torsion-free abelian group G, such that G/C is homogeneous completely decomposable of type f, and all elements of G\C are of type r, then G is a direct summand of G. Because of its numerous applications, many attempts have been made to extend the last result to situations in which G/C is not completely decomposable. Arnold and Lady succeeded in 1975 in the case that G is torsion-free of finite rank. Before we can state their result, we introduce some additional notation: Suppose that A and G are abelian groups. -
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. -
Algebraic Number Theory
Algebraic Number Theory Sergey Shpectorov January{March, 2010 This course in on algebraic number theory. This means studying problems from number theory with methods from abstract algebra. For a long time the main motivation behind the development of algebraic number theory was the Fermat Last Theorem. Proven in 1995 by Wiles with the help of Taylor, this theorem states that there are no positive integers x, y and z satisfying the equation xn + yn = zn; where n ≥ 3 is an integer. The proof of this statement for the particular case n = 4 goes back to Fibonacci, who lived four hundred years before Fermat. Modulo Fibonacci's result, Fermat Last Theorem needs to be proven only for the cases where n = p is an odd prime. By the end of the course we will hopefully see, as an application of our theory, how to prove the Fermat Last Theorem for the so-called regular primes. The idea of this belongs to Kummer, although we will, of course, use more modern notation and methods. Another accepted definition of algebraic number theory is that it studies the so-called number fields, which are the finite extensions of the field of ra- tional numbers Q. We mention right away, however, that most of this theory applies also in the second important case, known as the case of function fields. For example, finite extensions of the field of complex rational functions C(x) are function fields. We will stress the similarities and differences between the two types of fields, as appropriate. Finite extensions of Q are algebraic, and this ties algebraic number the- ory with Galois theory, which is an important prerequisite for us. -
Orders on Computable Torsion-Free Abelian Groups
Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z. -
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. -
Strongly Homogeneous Torsion Free Abelian Groups of Finite Rank
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 56, April 1976 STRONGLY HOMOGENEOUS TORSION FREE ABELIAN GROUPS OF FINITE RANK Abstract. An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of Q, the additive group of rationals. If G is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of G into a direct sum of indecompos- able subgroups are equivalent. D. K. Harrison, in an unpublished note, defined a p-special group to be a strongly homogeneous group such that G/pG » Z/pZ for some prime p and qG = G for all primes q =£ p and characterized these groups as the additive groups of certain valuation rings in algebraic number fields. Rich- man [7] provided a global version of this result. Call G special if G is strongly homogeneous, G/pG = 0 or Z/pZ for all primes p, and G contains a pure rank 1 subgroup isomorphic to a subring of Q. Special groups are then characterized as additive subgroups of the intersection of certain valuation rings in an algebraic number field (also see Murley [5]). Strongly homoge- neous groups of rank 2 are characterized in [2]. All of the above-mentioned characterizations can be derived from the more general (notation and terminology are as in Fuchs [3]): Theorem 1. -
Covering Group Theory for Compact Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 161 (2001) 255–267 www.elsevier.com/locate/jpaa Covering group theory for compact groups Valera Berestovskiia, Conrad Plautb;∗ aDepartment of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77, 644077, Russia bDepartment of Mathematics, University of Tennessee, Knoxville, TN 37919, USA Received 27 May 1999; received in revised form 27 March 2000 Communicated by A. Carboni Abstract We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism 2 K : G → G in this category. The abelian topological group 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary: 22C05; 22B05; secondary: 22KXX 1. Introduction and main results In [1] we developed a covering group theory and generalized notion of fundamental group that discards such traditional notions as local arcwise connectivity and local simple connectivity. The theory applies to a large category of “coverable” topological groups that includes, for example, all metrizable, connected, locally connected groups and even some totally disconnected groups. However, for locally compact groups, being coverable is equivalent to being arcwise connected [2]. Therefore, many connected compact groups (e.g. solenoids or the character group of ZN ) are not coverable. -
Algebra I Homework Three
Algebra I Homework Three Name: Instruction: In the following questions, you should work out the solutions in a clear and concise manner. Three questions will be randomly selected and checked for correctness; they count 50% grades of this homework set. The other questions will be checked for completeness; they count the rest 50% grades of the homework set. Staple this sheet of paper as the cover page of your homework set. 1. (Section 2.1) A subset X of an abelian group F is said to be linearly independent if n1x1 + ··· + nrxr = 0 always implies ni = 0 for all i (where ni 2 Z and x1; ··· ; xk are distinct elements of X). (a) X is linearly independent if and only if every nonzero element of the subgroup hXi may be written uniquely in the form n1x1 + ··· + nkxk (ni 2 Z, ni 6= 0, x1; ··· ; xk distinct elements of X). The set S := fnlx1 + ··· + nkxk j ni 2 Z; x1; ··· ; xk 2 X; k 2 Ng forms a subgroup of F and it contains X. So hXi ≤ S. Conversely, Every element of S is in hXi. Thus hXi = S. If X is linearly independent, suppose on the contrary, an element a 2 hXi can be expressed as two different linear combinations of elements in X: 0 0 0 0 a = nlx1 + ··· + nkxk = nlx1 + ··· + nkxk; x1; ··· ; xk 2 X; ni; ni 2 Z; ni 6= 0 or ni 6= 0 for i = 1; ··· ; k: 0 0 0 0 0 Then (n1 − n1)x1 + ··· + (nk − nk)xk = 0, and at least one of ni − ni is non-zero (since (n1; ··· ; nk) 6= (n1; ··· ; nk)). -
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.