The Group of Automorphisms of a 3-Generated 2-Group of Intermediate Growth
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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. -
On Just-Infiniteness of Locally Finite Groups and Their -Algebras
Bull. Math. Sci. DOI 10.1007/s13373-016-0091-4 On just-infiniteness of locally finite groups and their C∗-algebras V. Belyaev1 · R. Grigorchuk2 · P. Shumyatsky3 Received: 14 August 2016 / Accepted: 23 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We give a construction of a family of locally finite residually finite groups with just-infinite C∗-algebra. This answers a question from Grigorchuk et al. (Just- infinite C∗-algebras. https://arxiv.org/abs/1604.08774, 2016). Additionally, we show that residually finite groups of finite exponent are never just-infinite. Keywords Just infinite groups · C*-algebras Mathematics Subject Classification 46L05 · 20F50 · 16S34 For Alex Lubotzky on his 60th birthday. Communicated by Efim Zelmanov. R. Grigorchuk was supported by NSA Grant H98230-15-1-0328. P. Shumyatsky was supported by FAPDF and CNPq. B P. Shumyatsky [email protected] V. Belyaev [email protected] R. Grigorchuk [email protected] 1 Institute of Mathematics and Mechanics, S. Kovalevskaja 16, Ekaterinburg, Russia 2 Department of Mathematics, Texas A&M University, College Station, TX, USA 3 Department of Mathematics, University of Brasília, 70910 Brasília, DF, Brazil 123 V. Belyaev et al. 1 Introduction A group is called just-infinite if it is infinite but every proper quotient is finite. Any infi- nite finitely generated group has just-infinite quotient. Therefore any question about existence of an infinite finitely generated group with certain property which is pre- served under homomorphic images can be reduced to a similar question in the class of just-infinite groups. -
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. -
Arxiv:1510.00545V4 [Math.GR] 29 Jun 2016 7
SCHREIER GRAPHS OF GRIGORCHUK'S GROUP AND A SUBSHIFT ASSOCIATED TO A NON-PRIMITIVE SUBSTITUTION ROSTISLAV GRIGORCHUK, DANIEL LENZ, AND TATIANA NAGNIBEDA Abstract. There is a recently discovered connection between the spectral theory of Schr¨o- dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators. Contents Introduction 2 1. Subshifts and aperiodic order in one dimension 3 2. Schr¨odingeroperators with aperiodic order 5 2.1. Constancy of the spectrum and the integrated density of states (IDS) 5 2.2. The spectrum as a set and the absolute continuity of spectral measures 9 2.3. Aperiodic order and discrete random Schr¨odingeroperators 10 3. The substitution τ, its finite words Subτ and its subshift (Ωτ ;T ) 11 3.1. The substitution τ and its subshift: basic features 11 3.2. The main ingredient for our further analysis: n-partition and n-decomposition 14 3.3. The maximal equicontinuous factor of the dynamical system (Ωτ ;T ) 15 3.4. Powers and the index (critical exponent) of Subτ 19 3.5. -
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. -
What Is the Grigorchuk Group?
WHAT IS THE GRIGORCHUK GROUP? JAKE HURYN Abstract. The Grigorchuk group was first constructed in 1980 by Rostislav Grigorchuk, defined as a set of measure-preserving maps on the unit interval. In this talk a simpler construction in terms of binary trees will be given, and some important properties of this group will be explored. In particular, we study it as a negative example to a variant of the Burnside problem posed in 1902, an example of a non-linear group, and the first discovered example of a group of intermediate growth. 1. The Infinite Binary Tree We will denote the infinite binary tree by T . The vertex set of T is all finite words in the alphabet f0; 1g, and two words have an edge between them if and only if deleting the rightmost letter of one of them yields the other: ? 0 1 00 01 10 11 . Figure 1. The infinite binary tree T . This will in fact be a rooted tree, with the root at the empty sequence, so that any automorphism of T must fix the empty sequence. Here are some exercises about Aut(T ), the group of automorphisms of T . In this paper, exercises are marked with either ∗ or ∗∗ based on difficulty. Most are taken from exercises or statements made in the referenced books. Exercise 1. Show that the group Aut(T ) is uncountable and has a subgroup iso- morphic to Aut(T ) × Aut(T ).∗ Exercise 2. Impose a topology on Aut(T ) to make it into a topological group home- omorphic to the Cantor set.∗∗ Exercise 3. -
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. -
The Commensurability Relation for Finitely Generated Groups
J. Group Theory 12 (2009), 901–909 Journal of Group Theory DOI 10.1515/JGT.2009.022 ( de Gruyter 2009 The commensurability relation for finitely generated groups Simon Thomas (Communicated by A. V. Borovik) Abstract. There does not exist a Borel way of selecting an isomorphism class within each com- mensurability class of finitely generated groups. 1 Introduction Two groups G1, G2 are (abstractly) commensurable, written G1 QC G2, if and only if there exist subgroups Hi c Gi of finite index such that H1 G H2. In this paper, we shall consider the commensurability relation on the space G of finitely generated groups. Here G denotes the Polish space of finitely generated groups introduced by Grigorchuk [4]; i.e., the elements of G are the isomorphism types of marked groups 3G; c4, where G is a finitely generated group and c is a finite sequence of generators. (For a clear account of the basic properties of the space G, see either Champetier [1] or Grigorchuk [6].) Our starting point is the following result, which intuitively says that the isomorphism and commensurability relations on G have exactly the same complexity. Theorem 1.1. The isomorphism relation G and commensurability relation QC on the space G of finitely generated groups are Borel bireducible. In particular, there exists a Borel reduction from QC to G; i.e., a Borel map f : G ! G such that if G, H a G, then G QC H if and only if f ðGÞ G f ðHÞ: However, while the proof of Theorem 1.1 shows that there exists a Borel reduction from QC to G, it does not provide an explicit example of such a reduction. -
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 . -
Groups Acting on 1-Dimensional Spaces and Some Questions
GROUPS ACTING ON 1-DIMENSIONAL SPACES AND SOME QUESTIONS Andr´esNavas Universidad de Santiago de Chile ICM Special Session \Dynamical Systems and Ordinary Differential Equations" Rio de Janeiro, August 2018 A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). -
A Survey on Automorphism Groups of Finite P-Groups
A Survey on Automorphism Groups of Finite p-Groups Geir T. Helleloid Department of Mathematics, Bldg. 380 Stanford University Stanford, CA 94305-2125 [email protected] February 2, 2008 Abstract This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail. 1 Introduction The goal of this survey is to communicate some of what is known about the automorphism groups of finite p-groups. The focus is on three topics: explicit computations for familiar finite p-groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. Section 2 begins with some general theorems on automorphisms of finite p-groups. Section 3 continues with explicit examples of automorphism groups of finite p-groups found in the literature. This arXiv:math/0610294v2 [math.GR] 25 Oct 2006 includes the computations on the automorphism groups of the extraspecial p- groups (by Winter [65]), the Sylow p-subgroups of the Chevalley groups (by Gibbs [22] and others), the Sylow p-subgroups of the symmetric group (by Bon- darchuk [8] and Lentoudis [40]), and some p-groups of maximal class and related p-groups. -
THE GRIGORCHUK GROUP Contents Introduction 1 1. Preliminaries 2 2
THE GRIGORCHUK GROUP KATIE WADDLE Abstract. In this survey we will define the Grigorchuk group and prove some of its properties. We will show that the Grigorchuk group is finitely generated but infinite. We will also show that the Grigorchuk group is a 2-group, meaning that every element has finite order a power of two. This, along with Burnside's Theorem, gives that the Grigorchuk group is not linear. Contents Introduction 1 1. Preliminaries 2 2. The Definition of the Grigorchuk Group 5 3. The Grigorchuk Group is a 2-group 8 References 10 Introduction In a 1980 paper [2] Rostislav Grigorchuk constructed the Grigorchuk group, also known as the first Grigorchuk group. In 1984 [3] the group was proved by Grigorchuk to have intermediate word growth. This was the first finitely generated group proven to show such growth, answering the question posed by John Milnor [5] of whether such a group existed. The Grigorchuk group is one of the most important examples in geometric group theory as it exhibits a number of other interesting properties as well, including amenability and the characteristic of being just-infinite. In this paper I will prove some basic facts about the Grigorchuk group. The Grigorchuk group is a subgroup of the automorphism group of the binary tree T , which we will call Aut(T ). Section 1 will explore Aut(T ). The group Aut(T ) does not share all of its properties with the Grigorchuk group. For example we will prove: Proposition 0.1. Aut(T ) is uncountable. The Grigorchuk group, being finitely generated, cannot be uncountable.