DOCSLIB.ORG

An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L Advanced Lectures in Mathematics Volume XV An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L. Griess, Jr. International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XV An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L. Griess, Jr. 2010 Mathematics Subject Classification. 11H56, 20C10, 20C34, 20D08. Copyright © 2011 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with article.) In such cases, requests for permission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-206-0 Printed in the United States of America. 15 14 13 12 2 3 4 5 6 7 8 9 ADVANCED LECTURES IN MATHEMATICS Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Zhongshan University Guangzhou, China Contents 1 Introduction ................................................... 1 1.1 Outlineofthebook.......................................... 2 1.2 Suggestionsforfurtherreading................................ 3 1.3 Notations, background, conventions . 5 2 Bilinear Forms, Quadratic Forms and Their Isometry Groups .. 7 2.1 Standard results on quadratic forms and reflections, I . 9 2.1.1 Principalidealdomains(PIDs).......................... 10 2.2 Linearalgebra .............................................. 11 2.2.1 Interpretationofnonsingularity......................... 11 2.2.2 Extensionofscalars ................................... 13 2.2.3 Cyclicity of the values of a rational bilinear form . 13 2.2.4 Grammatrix ......................................... 14 2.3 Discriminantgroup.......................................... 16 2.4 Relationsbetweenalatticeandsublattices...................... 18 2.5 Involutionsonquadraticspaces ............................... 19 2.6 Standard results on quadratic forms and reflections, II . 20 2.6.1 Involutionsonlattices.................................. 20 2.7 Scaledisometries:normdoublersandtriplers ................... 23 3 General Results on Finite Groups and Invariant Lattices ...... 25 3.1 Discretenessofrationallattices................................ 25 3.2 Finitenessoftheisometrygroup............................... 25 3.3 Construction of a G-invariant bilinear form . 26 3.4 Semidirectproductsandwreathproducts....................... 27 3.5 Orthogonal decomposition of lattices . 28 4 Root Lattices of Types A, D, E ................................ 31 4.1 BackgroundfromLietheory.................................. 31 4.2 Rootlattices,theirdualsandtheirisometrygroups.............. 32 4.2.1 Definition of the An lattices ............................ 33 ii Contents 4.2.2 Definition of the Dn lattices............................ 34 4.2.3 Definition of the En lattices ............................ 34 4.2.4 Analysis of the An rootlattices ......................... 34 4.2.5 Analysis of the Dn rootlattices......................... 37 4.2.6 More on the isometry groups of type Dn ................. 39 4.2.7 Analysis of the En rootlattices ......................... 41 5 Hermite and Minkowski Functions ............................. 49 5.1 Smallranksandsmalldeterminants............................ 51 5.1.1 TablefortheMinkowskiandHermitefunctions ........... 52 5.1.2 Classifications of small rank, small determinant lattices . 53 5.2 Uniqueness of the lattices E6,E7 and E8 ....................... 54 5.3 Moresmallranksandsmalldeterminants....................... 57 6 Constructions of Lattices by Use of Codes ..................... 61 6.1 Definitionsandbasicresults.................................. 61 6.1.1 A construction of the E8-lattice with the binary [8, 4, 4] code 62 6.1.2 A construction of the E8-lattice with the ternary [4, 2, 3] code 64 6.2 Theproofs.................................................. 64 6.2.1 About power sets, boolean sums and quadratic forms . 64 6.2.2 Uniqueness of the binary [8, 4, 4]code.................... 65 6.2.3 Reed-Mullercodes..................................... 66 6.2.4 Uniquenessofthetetracode............................. 67 6.2.5 Theautomorphismgroupofthetetracode................ 67 6.2.6 Another characterization of [8, 4, 4]2 ..................... 69 6.2.7 Uniqueness of the E8-lattice implies uniqueness of the binary [8, 4, 4]code.................................... 69 6.3 Codes over F7 and a (mod 7)-construction of E8 ................. 70 6.3.1 The A6-lattice........................................ 71 7 Group Theory and Representations ............................ 73 7.1 Finitegroups ............................................... 73 7.2 Extraspecial p-groups........................................ 75 7.2.1 Extraspecialgroupsandcentralproducts................. 75 7.2.2 A normal form in an extraspecial group . 77 7.2.3 Aclassificationofextraspecialgroups.................... 77 7.2.4 An application to automorphism groups of extraspecial groups............................................... 79 7.3 Grouprepresentations........................................ 79 7.3.1 Representations of extraspecial p-groups.................. 80 7.3.2 ConstructionoftheBRWgroups........................ 82 7.3.3 Tensorproducts....................................... 85 7.4 Representation of the BRW group G ........................... 86 7.4.1 BRWgroupsasgroupextensions........................ 88 Contents iii 8 Overview of the Barnes-Wall Lattices .......................... 91 8.1 Somepropertiesoftheseries.................................. 91 8.2 Commutatordensity......................................... 93 8.2.1 Equivalence of 2/4-, 3/4-generation and commutator density for Dih8 ...................................... 93 8.2.2 Extraspecial groups and commutator density . 96 9 Construction and Properties of the Barnes-Wall Lattices ...... 99 9.1 TheBarnes-Wallseriesandtheirminimalvectors................ 99 9.2 UniquenessfortheBWlattices................................101 9.3 PropertiesoftheBRWgroups ................................102 9.4 Applicationstocodingtheory.................................103 9.5 Moreaboutminimumvectors.................................104 10 Even unimodular lattices in small dimensions ..................107 10.1Classificationsofevenunimodularlattices......................107 10.2ConstructionsofsomeNiemeierlattices........................108 10.2.1ConstructionofaLeechlattice..........................109 10.3BasictheoryoftheGolaycode................................111 10.3.1CharacterizationofcertainReed-Mullercodes.............111 10.3.2AbouttheGolaycode .................................112 10.3.3TheoctadTriangleanddodecads........................113 10.3.4AuniquenesstheoremfortheGolaycode.................116 10.4MinimalvectorsintheLeechlattice ...........................116 10.5FirstproofofuniquenessoftheLeechlattice....................117 10.6InitialresultsabouttheLeechlattice ..........................118 10.6.1 An automorphism which moves the standard frame . 118 10.7Turyn-styleconstructionofaLeechlattice......................119 10.8Equivariantunimodularizationsofevenlattices..................121 11 Pieces of Eight ................................................125 11.1Leechtriosandoverlattices...................................125 11.2 The order of the group O(Λ)..................................128 11.3 The simplicity of M24 ........................................130 11.4 Sublattices of Leech and subgroups of the isometry group . 132 11.5InvolutionsontheLeechlattice ...............................134 References .........................................................137 Index ..............................................................143 Appendix A The Finite Simple Groups...........................149 iv Contents Appendix B Reprints of Selected Articles ........................153 B.1 Pieces of Eight: Semiselfdual Lattices and a New Foundation for theTheoryofConwayandMathieuGroups.....................155 B.2 Pieces of 2d: Existence and Uniqueness for Barnes-Wall and YpsilantiLattices............................................181 B.3 Involutions on the Barnes-Wall Lattices
Load more

Recommended publications
  • ORTHOGONAL GROUP of CERTAIN INDEFINITE LATTICE Chang Heon Kim* 1. Introduction Given an Even Lattice M in a Real Quadratic Space
    JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No. 1, March 2007 ORTHOGONAL GROUP OF CERTAIN INDEFINITE LATTICE Chang Heon Kim* Abstract. We compute the special orthogonal group of certain lattice of signature (2; 1). 1. Introduction Given an even lattice M in a real quadratic space of signature (2; n), Borcherds lifting [1] gives a multiplicative correspondence between vec- tor valued modular forms F of weight 1¡n=2 with values in C[M 0=M] (= the group ring of M 0=M) and meromorphic modular forms on complex 0 varieties (O(2) £ O(n))nO(2; n)=Aut(M; F ). Here NM denotes the dual lattice of M, O(2; n) is the orthogonal group of M R and Aut(M; F ) is the subgroup of Aut(M) leaving the form F stable under the natural action of Aut(M) on M 0=M. In particular, if the signature of M is (2; 1), then O(2; 1) ¼ H: O(2) £ O(1) and Borcherds' theory gives a lifting of vector valued modular form of weight 1=2 to usual one variable modular form on Aut(M; F ). In this sense in order to work out Borcherds lifting it is important to ¯nd appropriate lattice on which our wanted modular group acts. In this article we will show: Theorem 1.1. Let M be a 3-dimensional even lattice of all 2 £ 2 integral symmetric matrices, that is, ½µ ¶ ¾ AB M = j A; B; C 2 Z BC Received December 30, 2006. 2000 Mathematics Subject Classi¯cation: Primary 11F03, 11H56.
    [Show full text]
  • ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆.
    [Show full text]
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
    [Show full text]
  • INTEGER POINTS and THEIR ORTHOGONAL LATTICES 2 to Remove the Congruence Condition
    INTEGER POINTS ON SPHERES AND THEIR ORTHOGONAL LATTICES MENNY AKA, MANFRED EINSIEDLER, AND URI SHAPIRA (WITH AN APPENDIX BY RUIXIANG ZHANG) Abstract. Linnik proved in the late 1950’s the equidistribution of in- teger points on large spheres under a congruence condition. The congru- ence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. 1. Introduction A theorem of Legendre, whose complete proof was given by Gauss in [Gau86], asserts that an integer D can be written as a sum of three squares if and only if D is not of the form 4m(8k + 7) for some m, k N. Let D = D N : D 0, 4, 7 mod8 and Z3 be the set of primitive∈ vectors { ∈ 6≡ } prim in Z3. Legendre’s Theorem also implies that the set 2 def 3 2 S (D) = v Zprim : v 2 = D n ∈ k k o is non-empty if and only if D D. This important result has been refined in many ways. We are interested∈ in the refinement known as Linnik’s problem. Let S2 def= x R3 : x = 1 . For a subset S of rational odd primes we ∈ k k2 set 2 D(S)= D D : for all p S, D mod p F× .
    [Show full text]
  • [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.
    [Show full text]
  • Groups with Identical Subgroup Lattices in All Powers
    GROUPS WITH IDENTICAL SUBGROUP LATTICES IN ALL POWERS KEITH A. KEARNES AND AGNES´ SZENDREI Abstract. Suppose that G and H are groups with cyclic Sylow subgroups. We show that if there is an isomorphism λ2 : Sub (G × G) ! Sub (H × H), then there k k are isomorphisms λk : Sub (G ) ! Sub (H ) for all k. But this is not enough to force G to be isomorphic to H, for we also show that for any positive integer N there are pairwise nonisomorphic groups G1; : : : ; GN defined on the same finite set, k k all with cyclic Sylow subgroups, such that Sub (Gi ) = Sub (Gj ) for all i; j; k. 1. Introduction To what extent is a finite group determined by the subgroup lattices of its finite direct powers? Reinhold Baer proved results in 1939 implying that an abelian group G is determined up to isomorphism by Sub (G3) (cf. [1]). Michio Suzuki proved in 1951 that a finite simple group G is determined up to isomorphism by Sub (G2) (cf. [10]). Roland Schmidt proved in 1981 that if G is a finite, perfect, centerless group, then it is determined up to isomorphism by Sub (G2) (cf. [6]). Later, Schmidt proved in [7] that if G has an elementary abelian Hall normal subgroup that equals its own centralizer, then G is determined up to isomorphism by Sub (G3). It has long been open whether every finite group G is determined up to isomorphism by Sub (G3). (For more information on this problem, see the books [8, 11].) One may ask more generally to what extent a finite algebraic structure (or algebra) is determined by the subalgebra lattices of its finite direct powers.
    [Show full text]
  • The Sphere Packing Problem in Dimension 8 Arxiv:1603.04246V2
    The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 8 In this paper we prove that no packing of unit balls in Euclidean space R has density greater than that of the E8-lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space d can be covered by non-overlapping unit balls. More precisely, let R be the Euclidean d vector space equipped with distance k · k and Lebesgue measure Vol(·). For x 2 R and d r 2 R>0 we denote by Bd(x; r) the open ball in R with center x and radius r. Let d X ⊂ R be a discrete set of points such that kx − yk ≥ 2 for any distinct x; y 2 X. Then the union [ P = Bd(x; 1) x2X d is a sphere packing. If X is a lattice in R then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P\ Bd(0; r)) ∆P (r) := ; r > 0: Vol(Bd(0; r)) We define the density of a packing P as the limit superior ∆P := lim sup ∆P (r): r!1 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 The number be want to know is the supremum over all possible packing densities ∆d := sup ∆P ; d P⊂R sphere packing 1 called the sphere packing constant. For which dimensions do we know the exact value of ∆d? Trivially, in dimension 1 we have ∆1 = 1.
    [Show full text]
  • Sphere Packing, Lattice Packing, and Related Problems
    Sphere packing, lattice packing, and related problems Abhinav Kumar Stony Brook April 25, 2018 Sphere packings Definition n A sphere packing in R is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler's conjecture, now a theorem of Hales and collaborators. mmm m mmm m There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. Face centered cubic packing Image: Greg A L (Wikipedia), CC BY-SA 3.0 license But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered.
    [Show full text]
  • Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
    Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A.
    [Show full text]
  • Physical Interpretation of the 30 8-Simplexes in the E8 240-Polytope
    Physical Interpretation of the 30 8-simplexes in the E8 240-Polytope: Frank Dodd (Tony) Smith, Jr. 2017 - viXra 1702.0058 248-dim Lie Group E8 has 240 Root Vectors arranged on a 7-sphere S7 in 8-dim space. The 12 vertices of a cuboctahedron live on a 2-sphere S2 in 3-dim space. They are also the 4x3 = 12 outer vertices of 4 tetrahedra (3-simplexes) that share one inner vertex at the center of the cuboctahedron. This paper explores how the 240 vertices of the E8 Polytope in 8-dim space are related to the 30x8 = 240 outer vertices (red in figure below) of 30 8-simplexes whose 9th vertex is a shared inner vertex (yellow in figure below) at the center of the E8 Polytope. The 8-simplex has 9 vertices, 36 edges, 84 triangles, 126 tetrahedron cells, 126 4-simplex faces, 84 5-simplex faces, 36 6-simplex faces, 9 7-simplex faces, and 1 8-dim volume The real 4_21 Witting polytope of the E8 lattice in R8 has 240 vertices; 6,720 edges; 60,480 triangular faces; 241,920 tetrahedra; 483,840 4-simplexes; 483,840 5-simplexes 4_00; 138,240 + 69,120 6-simplexes 4_10 and 4_01; and 17,280 = 2,160x8 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11. The cuboctahedron corresponds by Jitterbug Transformation to the icosahedron. The 20 2-dim faces of an icosahedon in 3-dim space (image from spacesymmetrystructure.wordpress.com) are also the 20 outer faces of 20 not-exactly-regular-in-3-dim tetrahedra (3-simplexes) that share one inner vertex at the center of the icosahedron, but that correspondence does not extend to the case of 8-simplexes in an E8 polytope, whose faces are both 7-simplexes and 7-cross-polytopes, similar to the cubocahedron, but not its Jitterbug-transform icosahedron with only triangle = 2-simplex faces.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:2010.10200V3 [Math.GT] 1 Mar 2021 We Deduce in Particular the Following
    HYPERBOLIC MANIFOLDS THAT FIBER ALGEBRAICALLY UP TO DIMENSION 8 GIOVANNI ITALIANO, BRUNO MARTELLI, AND MATTEO MIGLIORINI Abstract. We construct some cusped finite-volume hyperbolic n-manifolds Mn that fiber algebraically in all the dimensions 5 ≤ n ≤ 8. That is, there is a surjective homomorphism π1(Mn) ! Z with finitely generated kernel. The kernel is also finitely presented in the dimensions n = 7; 8, and this leads to the first examples of hyperbolic n-manifolds Mfn whose fundamental group is finitely presented but not of finite type. These n-manifolds Mfn have infinitely many cusps of maximal rank and hence infinite Betti number bn−1. They cover the finite-volume manifold Mn. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P5;:::;P8, and then applying some arguments of Jankiewicz { Norin { Wise [15] and Bestvina { Brady [6]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to Pn, and the algebra of integral octonions for the crucial dimensions n = 7; 8. Introduction We prove here the following theorem. Every hyperbolic manifold in this paper is tacitly assumed to be connected, complete, and orientable. Theorem 1. In every dimension 5 ≤ n ≤ 8 there are a finite volume hyperbolic 1 n-manifold Mn and a map f : Mn ! S such that f∗ : π1(Mn) ! Z is surjective n with finitely generated kernel. The cover Mfn = H =ker f∗ has infinitely many cusps of maximal rank. When n = 7; 8 the kernel is also finitely presented. arXiv:2010.10200v3 [math.GT] 1 Mar 2021 We deduce in particular the following.
    [Show full text]