Automorphism Groups and Their Actions
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On Automorphisms and Endomorphisms of Projective Varieties
On automorphisms and endomorphisms of projective varieties Michel Brion Abstract We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal pro- jective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X). Key words: automorphism group scheme, endomorphism semigroup scheme MSC classes: 14J50, 14L30, 20M20 1 Introduction and statement of the results By a result of Winkelmann (see [22]), every connected real Lie group G can be realized as the automorphism group of some complex Stein manifold X, which may be chosen complete, and hyperbolic in the sense of Kobayashi. Subsequently, Kan showed in [12] that we may further assume dimC(X) = dimR(G). We shall obtain a somewhat similar result for connected algebraic groups. We first introduce some notation and conventions, and recall general results on automorphism group schemes. Throughout this article, we consider schemes and their morphisms over a fixed field k. Schemes are assumed to be separated; subschemes are locally closed unless mentioned otherwise. By a point of a scheme S, we mean a Michel Brion Institut Fourier, Universit´ede Grenoble B.P. 74, 38402 Saint-Martin d'H`eresCedex, France e-mail: [email protected] 1 2 Michel Brion T -valued point f : T ! S for some scheme T .A variety is a geometrically integral scheme of finite type. We shall use [17] as a general reference for group schemes. -
Orbits of Automorphism Groups of Fields
ORBITS OF AUTOMORPHISM GROUPS OF FIELDS KIRAN S. KEDLAYA AND BJORN POONEN Abstract. We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann \generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic. Can an infinite field K have so many automorphisms that the action of the automorphism group on the elements of K has only finitely many orbits? In Section 1, we prove that the answer is \no" (Theorem 1.1), even though the corresponding answer for division rings is \yes" (see Remark 1.2). Our proof constructs a \trace map" from the given field to a finite field, and exploits the peculiar combination of additive and multiplicative properties of this map. Section 2 attempts to prove a relative version of Theorem 1.1, by considering, for a non- trivial extension of fields k ⊂ K, the action of Aut(K=k) on K. In this situation each element of k forms an orbit, so we study only the orbits of Aut(K=k) on K − k. -
CURRICULUM VITAE PISIER Gilles Jean Georges Born November 18 1950 in Nouméa, New Caledonia. French Nationality 1966-67 Mathéma
CURRICULUM VITAE PISIER Gilles Jean Georges Born November 18 1950 in Noum´ea,New Caledonia. French Nationality 1966-67 Math´ematiquesEl´ementaires at Lyc´eeBuffon, Paris. 1967 Baccalaur´eat,section C. 1967-68 Math´ematiquesSup´erieuresand Math´ematiquesSp´eciales 1968-69 at Lyc´eeLouis-le Grand, Paris. 1969-72 Student at Ecole´ Polytechnique. 1971 Ma^ıtrisede math´ematiquesUniversit´ePARIS VII. 1972 D.E.A. de Math´ematiquespures Universit´ePARIS VI. Oct. 72 Stragiaire de Recherche at C.N.R.S. Oct. 74 Attach´ede Recherche at C.N.R.S. Nov. 77 Th`esede doctorat d'´etat `es-sciencesmath´ematiques, soutenue le 10 Novembre 1977 from Universit´ePARIS VII under the supervision of L. Schwartz. Oct. 79 Charg´ede Recherche at C.N.R.S. Oct. 81 Professor at the University of PARIS VI. Oct. 84-Jan.85 Visiting Professor at IHES (Bures s. Yvette). Sept. 85 Distinguished Professor (Owen Chair of Mathematics), Texas A&M University. Nov. 88 Short term Visitor, Institute of Advanced Study (Princeton). Feb. 90-Apr.90 Visiting Professor at IHES (Bures s. Yvette). Feb. 1991 Professeur de Classe Exceptionnelle (Universit´eParis 6) 1 VARIOUS DISTINCTIONS { Salem Prize 1979. { Cours Peccot at Coll`egede France 1981. { Prix Carri`erede l'Acad´emiedes Science de Paris 1982. { Invited speaker at the International Congress of Mathematicians (Warsaw,1983). {Fellow of the Institute of Mathematical Statistics 1989. {Grands prix de l'Acad´emiedes Sciences de Paris: Prix Fond´epar l'Etat 1992 {Invited one hour address at A.M.S. meeting in College Station, october 93. {Faculty Distinguished Achievement Award in Research 1993, Texas A&M University (from the Association of Former Students) {Elected \Membre correspondant" by \Acad´emiedes Sciences de Paris", April 94. -
The Pure Symmetric Automorphisms of a Free Group Form a Duality Group
THE PURE SYMMETRIC AUTOMORPHISMS OF A FREE GROUP FORM A DUALITY GROUP NOEL BRADY, JON MCCAMMOND, JOHN MEIER, AND ANDY MILLER Abstract. The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups. 1. Introduction Let Fn be a finite rank free group with fixed free basis X = fx1; : : : ; xng. The symmetric automorphism group of Fn, hereafter denoted Σn, consists of those au- tomorphisms that send each xi 2 X to a conjugate of some xj 2 X. The pure symmetric automorphism group, denoted PΣn, is the index n! subgroup of Σn of symmetric automorphisms that send each xi 2 X to a conjugate of itself. The quotient of PΣn by the inner automorphisms of Fn will be denoted OPΣn. In this note we prove: Theorem 1.1. The group OPΣn is a duality group of dimension n − 2. Corollary 1.2. The group PΣn is a duality group of dimension n − 1, hence Σn is a virtual duality group of dimension n − 1. (In fact we establish slightly more: the dualizing module in both cases is -free.) Corollary 1.2 follows immediately from Theorem 1.1 since Fn is a 1-dimensional duality group, there is a short exact sequence 1 ! Fn ! PΣn ! OPΣn ! 1 and any duality-by-duality group is a duality group whose dimension is the sum of the dimensions of its constituents (see Theorem 9.10 in [2]). That the virtual cohomological dimension of Σn is n − 1 was previously estab- lished by Collins in [9]. -
Generalized Quaternions
GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8. -
On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph. -
The Centralizer of a Group Automorphism
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 27-41 (1978) The Centralizer of a Group Automorphism CARLTON J. MAXSON AND KIRBY C. SWTH Texas A& M University, College Station, Texas 77843 Communicated by A. Fr6hlich Received May 31, 1971 Let G be a finite group. The structure of the near-ring C(A) of identity preserving functionsf : G + G, which commute with a given automorphism A of G, is investigated. The results are then applied to the case in which G is a finite vector space and A is an invertible linear transformation. Let A be a linear transformation on a finite-dimensional vector space V over a field F. The problem of determining the structure of the ring of linear trans- formations on V which commute with A has been studied extensively (e.g., [3, 5, 81). In this paper we consider a nonlinear analogue to this well-known problem which arises naturally in the study of automorphisms of a linear auto- maton [6]. Specifically let G be a finite group, written additively, and let A be an auto- morphism of G. If C(A) = {f: G + G j fA = Af and f (0) = 0 where 0 is the identity of G), then C(A) forms a near ring under the operations of pointwise addition and function composition. That is (C(A), +) is a group, (C(A), .) is a monoid, (f+-g)h=fh+gh for all f, g, hcC(A) and f.O=O.f=O, f~ C(A). -
Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups
Automorphism groups of free groups, surface groups and free abelian groups Martin R. Bridson and Karen Vogtmann The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the map- ± ping class groups Mod (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a success- ful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similar- ities and differences between these series of groups and present some open problems motivated by this philosophy. ± Similarities among the groups Out(Fn), GL(n, Z) and Mod (Sg) begin with the fact that these are the outer automorphism groups of the most prim- itive types of torsion-free discrete groups, namely free groups, free abelian groups and the fundamental groups of closed orientable surfaces π1Sg. In the ± case of Out(Fn) and GL(n, Z) this is obvious, in the case of Mod (Sg) it is a classical theorem of Nielsen. -
Isomorphisms and Automorphisms
Isomorphisms and Automorphisms Definition As usual an isomorphism is defined as a map between objects that preserves structure, for general designs this means: Suppose (X,A) and (Y,B) are two designs. They are said to be isomorphic if there exists a bijection α: X → Y such that if we apply α to the elements of any block of A we obtain a block of B and all blocks of B are obtained this way. Symbolically we can write this as: B = [ {α(x) | x ϵ A} : A ϵ A ]. The bijection α is called an isomorphism. Example We've seen two versions of a (7,4,2) symmetric design: k = 4: {3,5,6,7} {4,6,7,1} {5,7,1,2} {6,1,2,3} {7,2,3,4} {1,3,4,5} {2,4,5,6} Blocks: y = [1,2,3,4] {1,2} = [1,2,5,6] {1,3} = [1,3,6,7] {1,4} = [1,4,5,7] {2,3} = [2,3,5,7] {2,4} = [2,4,6,7] {3,4} = [3,4,5,6] α:= (2576) [1,2,3,4] → {1,5,3,4} [1,2,5,6] → {1,5,7,2} [1,3,6,7] → {1,3,2,6} [1,4,5,7] → {1,4,7,2} [2,3,5,7] → {5,3,7,6} [2,4,6,7] → {5,4,2,6} [3,4,5,6] → {3,4,7,2} Incidence Matrices Since the incidence matrix is equivalent to the design there should be a relationship between matrices of isomorphic designs. -
Automorphisms of Group Extensions
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 155, Number t, March 1971 AUTOMORPHISMS OF GROUP EXTENSIONS BY CHARLES WELLS Abstract. If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j>of E which takes G onto itself induces automorphisms t on G and a on n. However, for a pair (a, t) of automorphism of n and G, there may not be an automorphism of E inducing the pair. Let à: n —*■Out G be the homomorphism induced by the given extension. A pair (a, t) e Aut n x Aut G is called compatible if a fixes ker á, and the automorphism induced by a on Hü is the same as that induced by the inner automorphism of Out G determined by t. Let C< Aut IT x Aut G be the group of compatible pairs. Let Aut (E; G) denote the group of automorphisms of E fixing G. The main result of this paper is the construction of an exact sequence 1 -» Z&T1,ZG) -* Aut (E; G)-+C^ H*(l~l,ZG). The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense. 1. Notation. If G is a group with subgroup H, we write H<G; if H is normal in G, H<¡G. CGH and NGH are the centralizer and normalizer of H in G. Aut G, Inn G, Out G, and ZG are the automorphism group, the inner automorphism group, the outer automorphism group, and the center of G, respectively. -
Amenability, Group C -Algebras and Operator Spaces (WEP and LLP For
Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Gilles Pisier Texas A&M University Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Main Source... E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C∗-algebras. Invent. Math. 112 (1993), 449{489. Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) C ∗-algebras A C ∗-algebra is a closed self-adjoint subalgebra A ⊂ B(H) of the space of bounded operators on a Hilbert space H An operator space is a (closed) subspace E ⊂ A of a C ∗-algebra I will restrict to unital C ∗-algebras The norm on the ∗-algebra A satisfies 8x; y 2 A kxyk ≤ kxkkyk kxk = kx∗k kx∗xk = kxk2 Such norms are called C ∗-norms After completion (or if A is already complete): there is a unique C ∗-norm on A Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Then any such A can be written as A = span[π(G)] for some discrete group G and some unitary representation π : G ! B(H) of G on H Typical operator space E = span[π(S)] S ⊂ G Throughout I will restrict to discrete groups Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Tensor products On the algebraic tensor product (BEFORE completion) A ⊗ B there is a minimal and a maximal C ∗-norm denoted by k kmin and k kmax and in general k kmin ≤6= k kmax C ∗-norms 8x; y 2 A kxyk ≤ kxkkyk kxk = kx∗k kx∗xk = kxk2 Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Nuclear pairs Let A; B be C ∗-algebras Definition The pair (A; B) is said to be a nuclear pair if the minimal and maximal C ∗-norms coincide on the algebraic tensor product A ⊗ B, in other words A ⊗min B = A ⊗max B Definition A C ∗-algebra A is called nuclear if this holds for ANY B Gilles Pisier Amenability, Group C ∗-algebras and Operator spaces (WEP and LLP for C ∗(G)) Let A1 = span[π(G1)] ⊂ B(H1) A2 = span[π(G2)] ⊂ B(H2) for some unitary rep. -
Free and Linear Representations of Outer Automorphism Groups of Free Groups
Free and linear representations of outer automorphism groups of free groups Dawid Kielak Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2012 This thesis is dedicated to Magda Acknowledgements First and foremost the author wishes to thank his supervisor, Martin R. Bridson. The author also wishes to thank the following people: his family, for their constant support; David Craven, Cornelia Drutu, Marc Lackenby, for many a helpful conversation; his office mates. Abstract For various values of n and m we investigate homomorphisms Out(Fn) ! Out(Fm) and Out(Fn) ! GLm(K); i.e. the free and linear representations of Out(Fn) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(Fn) we prove that each ho- momorphism Out(Fn) ! GLm(K) factors through the natural map ∼ πn : Out(Fn) ! GL(H1(Fn; Z)) = GLn(Z) whenever n = 3; m < 7 and char(K) 62 f2; 3g, and whenever n + 1 n > 5; m < 2 and char(K) 62 f2; 3; : : : ; n + 1g: We also construct a new infinite family of linear representations of Out(Fn) (where n > 2), which do not factor through πn. When n is odd these have the smallest dimension among all known representations of Out(Fn) with this property. Using the above results we establish that the image of every homomor- phism Out(Fn) ! Out(Fm) is finite whenever n = 3 and n < m < 6, and n of cardinality at most 2 whenever n > 5 and n < m < 2 .