Automorphisms of Free Groups and Outer Space
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Math 602, Fall 2002. BIII (A). We First Prove the Following Proposition
Homework I (due September 30), Math 602, Fall 2002. BIII (a). We first prove the following proposition: Proposition 1.1 Given a group G, for any finite normal subgroup, H, of G and any p-Sylow subgroup P of H, we have G = NG(P )H. Proof . For every g ∈ G, clearly, g−1P g is a subgroup of g−1Hg = H, since H is normal in G. Since |g−1P g| = |P |, the subgroup g−1P g is also a p-Sylow subgroup of H. By Sylow II, g−1P g is some conjugate of P in H, i.e., g−1P g = hP h−1 for some h ∈ H. −1 −1 Thus, ghP h g = P , which implies that gh ∈ NG(P ), so g ∈ NG(P )H. Since the reasoning holds for every g ∈ G, we get G = NG(P )H. Now, since H is normal in G, by the second homomorphism theorem, we know that ∼ G/H = (NG(P )H)/H = NG(P )/(NG(P ) ∩ H). Moreover, it is clear that NG(P ) ∩ H = ∼ NH (P ), so G/H = NG(P )/NH (P ), as desired. (b) We shall prove that every p-Sylow subgroup of Φ(G) is normal in Φ(G) (and in fact, in G). From this, we will deduce that Φ(G) has property (N). Indeed, if we inspect the proof of the proposition proved in class stating that if G is a finite group that has (N), then G is isomorphic to the product of its p-Sylow subgroups, we see that this proof only depends on the fact that every p-Sylow subgroup of G is normal in G. -
Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a -
Arxiv:Math/0511624V1
Automorphism groups of polycyclic-by-finite groups and arithmetic groups Oliver Baues ∗ Fritz Grunewald † Mathematisches Institut II Mathematisches Institut Universit¨at Karlsruhe Heinrich-Heine-Universit¨at D-76128 Karlsruhe D-40225 D¨usseldorf May 18, 2005 Abstract We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homo- topy theory. 2000 Mathematics Subject Classification: Primary 20F28, 20G30; Secondary 11F06, 14L27, 20F16, 20F34, 22E40, 55P10 arXiv:math/0511624v1 [math.GR] 25 Nov 2005 ∗e-mail: [email protected] †e-mail: [email protected] 1 Contents 1 Introduction 4 1.1 Themainresults ........................... 4 1.2 Outlineoftheproofsandmoreresults . 6 1.3 Cohomologicalrepresentations . 8 1.4 Applications to the groups of homotopy self-equivalences of spaces 9 2 Prerequisites on linear algebraic groups and arithmetic groups 12 2.1 Thegeneraltheory .......................... 12 2.2 Algebraicgroupsofautomorphisms. 15 3 The group of automorphisms of a solvable-by-finite linear algebraic group 17 3.1 The algebraic structure of Auta(H)................ -
A CHARACTERISTIC SUBGROUP of a P-GROUP
Pacific Journal of Mathematics A CHARACTERISTIC SUBGROUP OF A p-GROUP CHARLES RAY HOBBY Vol. 10, No. 3 November 1960 A CHARACTERISTIC SUBGROUP OF A p-GROVP CHARLES HOBBY If x, y are elements and H, K subsets of the p-group G, we shall denote by [x, y] the element y~px~p(xy)p of G, and by [H, K] the sub- group of G generated by the set of all [h, k] for h in H and k in K. We call a p-group G p-abelίan if (xy)p = xpyp for all elements x, y of G. If we let Θ(G) — [G, G] then #(G) is a characteristic subgroup of G and Gjθ{G) is p-abelian. In fact, Θ(G) is the minimal normal subgroup N of G for which G/AΓ is p-abelian. It is clear that Θ(G) is contained in the derived group of G, and G/Θ(G) is regular in the sense of P. Hall [3] Theorem 1 lists some elementary properties of p-abelian groups. These properties are used to obtain a characterization of p-groups G (for p > 3) in which the subgroup generated by the pth powers of elements of G coincides with the Frattini subgroup of G (Theorems 2 and 3). A group G is said to be metacyclic if there exists a cyclic normal sub- group N with G/N cyclic. Theorem 4 states that a p-group G, for p > 2, is metacyclic if and only if Gjθ(G) is metacyclic. -
Algorithmic Constructions of Relative Train Track Maps and Cts Arxiv
Algorithmic constructions of relative train track maps and CTs Mark Feighn∗ Michael Handely June 30, 2018 Abstract Building on [BH92, BFH00], we proved in [FH11] that every element of the outer automorphism group of a finite rank free group is represented by a particularly useful relative train track map. In the case that is rotationless (every outer automorphism has a rotationless power), we showed that there is a type of relative train track map, called a CT, satisfying additional properties. The main result of this paper is that the constructions of these relative train tracks can be made algorithmic. A key step in our argument is proving that it 0 is algorithmic to check if an inclusion F @ F of φ-invariant free factor systems is reduced. We also give applications of the main result. Contents 1 Introduction 2 2 Relative train track maps in the general case 5 2.1 Some standard notation and definitions . .6 arXiv:1411.6302v4 [math.GR] 6 Jun 2017 2.2 Algorithmic proofs . .9 3 Rotationless iterates 13 3.1 More on markings . 13 3.2 Principal automorphisms and principal points . 14 3.3 A sufficient condition to be rotationless and a uniform bound . 16 3.4 Algorithmic Proof of Theorem 2.12 . 20 ∗This material is based upon work supported by the National Science Foundation under Grant No. DMS-1406167 and also under Grant No. DMS-14401040 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester. yThis material is based upon work supported by the National Science Foundation under Grant No. -
Curriculum Vitæ of Peter B. Shalen
Curriculum Vit½ of Peter B. Shalen Address Department of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 312-996-4825 (FAX) 312-996-1491 E-mail: [email protected] Education Ph.D. 1972 Harvard University B.A. 1966 Harvard College Pensionnaire ¶etranger, Ecole Normale Sup¶erieure, Paris, 1966-67 Employment 1985|present Professor University of Illinois at Chicago 1998-99 Long-term visitor, University of Chicago June, 1998 Professeur Invit¶e Universit¶e Paul Sabatier, Toulouse June, 1997 Professeur Invit¶e Universit¶e de Bourgogne June, 1996 Professeur Invit¶e University of Paris June, 1993 Professeur Invit¶e Universit¶e Paul Sabatier, Toulouse Spring 1985 Member Mathematical Sciences Research Institute, Berkeley Fall 1984 Professeur Associ¶e University of Paris (Orsay) Spring 1984 Professeur Associ¶e University of Nantes 1983|85 Professor Rice University 1981|82 Visiting Scholar Columbia University 1979|83 Associate Professor Rice University 1978|79 Visiting Member Courant Institute of Mathematical Sciences, N.Y.U. 1974|79 Assistant Professor Rice University 1971|74 J.F. Ritt Assistant Professor Columbia University 1 Professional Honors Alfred P. Sloan Foundation Fellowship for Basic Research, 1977|79. Invited one-hour address, A.M.S. Regional Meeting, University of Wisconsin at Parkside, October, 1980. J. Clarence Karcher Lectures in Mathematics, University of Oklahoma, April 1980. Member, Mathematical Sciences Research Institute, Berkeley, Spring 1985, and December 1988. Invited 45-minute address, International Congress of Mathematicians, Berkeley, California, August 1986. Fourth annual Zabrodsky lecture, Hebrew University, Jerusalem. December, 1990. University Scholar award, University of Illinois, 1996 Invited visits, American Institute of Mathematics, May 2000 and May 2002. -
14.2 Covers of Symmetric and Alternating Groups
Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates. -
Arxiv:1507.08163V2 [Math.DG] 10 Nov 2015 Ainld C´Ordoba)
GEOMETRIC FLOWS AND THEIR SOLITONS ON HOMOGENEOUS SPACES JORGE LAURET Dedicated to Sergio. Abstract. We develop a general approach to study geometric flows on homogeneous spaces. Our main tool will be a dynamical system defined on the variety of Lie algebras called the bracket flow, which coincides with the original geometric flow after a natural change of variables. The advantage of using this method relies on the fact that the possible pointed (or Cheeger-Gromov) limits of solutions, as well as self-similar solutions or soliton structures, can be much better visualized. The approach has already been worked out in the Ricci flow case and for general curvature flows of almost-hermitian structures on Lie groups. This paper is intended as an attempt to motivate the use of the method on homogeneous spaces for any flow of geometric structures under minimal natural assumptions. As a novel application, we find a closed G2-structure on a nilpotent Lie group which is an expanding soliton for the Laplacian flow and is not an eigenvector. Contents 1. Introduction 2 1.1. Geometric flows on homogeneous spaces 2 1.2. Bracket flow 3 1.3. Solitons 4 2. Some linear algebra related to geometric structures 5 3. The space of homogeneous spaces 8 3.1. Varying Lie brackets viewpoint 8 3.2. Invariant geometric structures 9 3.3. Degenerations and pinching conditions 11 3.4. Convergence 12 4. Geometric flows 13 arXiv:1507.08163v2 [math.DG] 10 Nov 2015 4.1. Bracket flow 14 4.2. Evolution of the bracket norm 18 4.3. -
Topics on the Geometry of Homogeneous Spaces
TOPICS ON THE GEOMETRY OF HOMOGENEOUS SPACES LAURENT MANIVEL Abstract. This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r^ole.This selection is largely arbitrary and mainly reflects the interests of the author. Rational homogeneous varieties are very special projective varieties, which ap- pear in a variety of circumstances as exhibiting extremal behavior. In the quite re- cent years, a series of very interesting examples of pairs (sometimes called Fourier- Mukai partners) of derived equivalent, but not isomorphic, and even non bira- tionally equivalent manifolds have been discovered by several authors, starting from the special geometry of certain homogeneous spaces. We will not discuss derived categories and will not describe these derived equivalences: this would require more sophisticated tools and much ampler discussions. Neither will we say much about Homological Projective Duality, which can be considered as the unifying thread of all these apparently disparate examples. Our much more modest goal will be to describe their geometry, starting from the ambient homogeneous spaces. In order to do so, we will have to explain how one can approach homogeneous spaces just playing with Dynkin diagram, without knowing much about Lie the- ory. In particular we will explain how to describe the VMRT (variety of minimal rational tangents) of a generalized Grassmannian. This will show us how to com- pute the index of these varieties, remind us of their importance in the classification problem of Fano manifolds, in rigidity questions, and also, will explain their close relationships with prehomogeneous vector spaces. -
WHAT IS Outer Space?
WHAT IS Outer Space? Karen Vogtmann To investigate the properties of a group G, it is with a metric of constant negative curvature, and often useful to realize G as a group of symmetries g : S ! X is a homeomorphism, called the mark- of some geometric object. For example, the clas- ing, which is well-defined up to isotopy. From this sical modular group P SL(2; Z) can be thought of point of view, the mapping class group (which as a group of isometries of the upper half-plane can be identified with Out(π1(S))) acts on (X; g) f(x; y) 2 R2j y > 0g equipped with the hyper- by composing the marking with a homeomor- bolic metric ds2 = (dx2 + dy2)=y2. The study of phism of S { the hyperbolic metric on X does P SL(2; Z) and its subgroups via this action has not change. By deforming the metric on X, on occupied legions of mathematicians for well over the other hand, we obtain a neighborhood of the a century. point (X; g) in Teichm¨uller space. We are interested here in the (outer) automor- phism group of a finitely-generated free group. u v Although free groups are the simplest and most fundamental class of infinite groups, their auto- u v u v u u morphism groups are remarkably complex, and v x w v many natural questions about them remain unan- w w swered. We will describe a geometric object On known as Outer space , which was introduced in w w u v [2] to study Out(Fn). -
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. -
THE CONJUGACY CLASS RANKS of M24 Communicated by Robert Turner Curtis 1. Introduction M24 Is the Largest Mathieu Sporadic Simple
International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 6 No. 4 (2017), pp. 53-58. ⃝c 2017 University of Isfahan ijgt.ui.ac.ir www.ui.ac.ir THE CONJUGACY CLASS RANKS OF M24 ZWELETHEMBA MPONO Dedicated to Professor Jamshid Moori on the occasion of his seventieth birthday Communicated by Robert Turner Curtis 10 3 Abstract. M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of M24 with respect to the conjugacy classes of all its nonidentity elements. 1. Introduction 10 3 M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. It is a 5-transitive permutation group on a set of 24 points such that: (i) the stabilizer of a point is M23 which is 4-transitive (ii) the stabilizer of two points is M22 which is 3-transitive (iii) the stabilizer of a dodecad is M12 which is 5-transitive (iv) the stabilizer of a dodecad and a point is M11 which is 4-transitive M24 has a trivial Schur multiplier, a trivial outer automorphism group and it is the automorphism group of the Steiner system of type S(5,8,24) which is used to describe the Leech lattice on which M24 acts. M24 has nine conjugacy classes of maximal subgroups which are listed in [4], its ordinary character table is found in [4], its blocks and its Brauer character tables corresponding to the various primes dividing its order are found in [9] and [10] respectively and its complete prime spectrum is given by 2; 3; 5; 7; 11; 23.