The Topology of Out(Fn)
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CERF THEORY for GRAPHS Allen Hatcher and Karen Vogtmann
CERF THEORY FOR GRAPHS Allen Hatcher and Karen Vogtmann ABSTRACT. We develop a deformation theory for k-parameter families of pointed marked graphs with xed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), compu- tations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn). §1. Introduction It is standard practice to study groups by studying their actions on well-chosen spaces. In the case of the outer automorphism group Out(Fn) of the free group Fn on n generators, a good space on which this group acts was constructed in [3]. In the present paper we study an analogous space on which the automorphism group Aut(Fn) acts. This space, which we call An, is a sort of Teichmuller space of nite basepointed graphs with fundamental group Fn. As often happens, specifying basepoints has certain technical advantages. In particu- lar, basepoints give rise to a nice ltration An,0 An,1 of An by degree, where we dene the degree of a nite basepointed graph to be the number of non-basepoint vertices, counted “with multiplicity” (see section 3). The main technical result of the paper, the Degree Theorem, implies that the pair (An, An,k)isk-connected, so that An,k behaves something like the k-skeleton of An. From the Degree Theorem we derive these applications: (1) We show the natural stabilization map Hi(Aut(Fn); Z) → Hi(Aut(Fn+1); Z)isan isomorphism for n 2i + 3, a signicant improvement on the quadratic dimension range obtained in [5]. -
Categories of Sets with a Group Action
Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. -
Stable Homology of Automorphism Groups of Free Groups
Annals of Mathematics 173 (2011), 705{768 doi: 10.4007/annals.2011.173.2.3 Stable homology of automorphism groups of free groups By Søren Galatius Abstract Homology of the group Aut(Fn) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular we confirm the conjecture that stable rational homology of Aut(Fn) vanishes. Contents 1. Introduction 706 1.1. Results 706 1.2. Outline of proof 708 1.3. Acknowledgements 710 2. The sheaf of graphs 710 2.1. Definitions 710 2.2. Point-set topological properties 714 3. Homotopy types of graph spaces 718 3.1. Graphs in compact sets 718 3.2. Spaces of graph embeddings 721 3.3. Abstract graphs 727 3.4. BOut(Fn) and the graph spectrum 730 4. The graph cobordism category 733 4.1. Poset model of the graph cobordism category 735 4.2. The positive boundary subcategory 743 5. Homotopy type of the graph spectrum 750 5.1. A pushout diagram 751 5.2. A homotopy colimit decomposition 755 5.3. Hatcher's splitting 762 6. Remarks on manifolds 764 References 766 S. Galatius was supported by NSF grants DMS-0505740 and DMS-0805843, and the Clay Mathematics Institute. 705 706 SØREN GALATIUS 1. Introduction 1.1. Results. Let F = x ; : : : ; x be the free group on n generators n h 1 ni and let Aut(Fn) be its automorphism group. -
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. -
Serre's Conjecture II: a Survey
Serre’s conjecture II: a survey Philippe Gille Dedicated to Parimala Abstract We provide a survey of Serre’s conjecture II (1962) on the vanishing of Galois cohomology for simply connected semisimple groups defined over a field of cohomological dimension 2. ≤ Math subject classification: 20G05. 1 Introduction Serre’s original conjecture II (1962) states that the Galois cohomology set H1(k,G) vanishes for a semisimple simply connected algebraic group G defined over a perfect field of cohomological dimension 2 [53, 4.1] [54, II.3.1]. This means that G- ≤ § torsors (or principal homogeneous spaces) over Spec(k) are trivial. For example, if A is a central simple algebra defined over a field k and c k , ∈ × the subvariety X := nrd(y)=c GL (A) c { }⊂ 1 of elements of reduced norm c is a torsor under the special linear group G = SL1(A) which is semisimple and simply connected. If cd(k) 2, we thus expect that this ≤ G-torsor is trivial, i.e., X (k) = /0.Applying this to all c k , we thus expect that c ∈ × the reduced norm map A k is surjective. × → × For imaginary number fields, the surjectivity of reduced norms goes back to Eich- ler in 1938 (see [40, 5.4]). For function fields of complex surfaces, this follows § from the Tsen-Lang theorem because the reduced norm is a homogeneous form of degree deg(A) in deg(A)2-indeterminates [54, II.4.5]. In the general case, the surjec- Philippe Gille DMA, UMR 8553 du CNRS, Ecole normale superieure,´ 45 rue d’Ulm, F-75005 Paris, France e-mail: [email protected] 37 38 Philippe Gille tivity of reduced norms is due to Merkurjev-Suslin in 1981 [60, Th. -
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). -
Cohomological Dimension of the Braid and Mapping Class Groups of Surfaces Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado
Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado To cite this version: Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado. Embeddings and the (virtual) cohomo- logical dimension of the braid and mapping class groups of surfaces. Confluentes Mathematici, Institut Camille Jordan et Unité de Mathématiques Pures et Appliquées, 2018, 10 (1), pp.41-61. hal-01377681 HAL Id: hal-01377681 https://hal.archives-ouvertes.fr/hal-01377681 Submitted on 11 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EMBEDDINGS AND THE (VIRTUAL) COHOMOLOGICAL DIMENSION OF THE BRAID AND MAPPING CLASS GROUPS OF SURFACES DACIBERG LIMA GONC¸ALVES, JOHN GUASCHI, AND MIGUEL MALDONADO Abstract. In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of [4, 14] to show that the mapping class group MCG(N; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. -
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. -
Ergodic Currents Dual to a Real Tree Thierry Coulbois, Arnaud Hilion
Ergodic currents dual to a real tree Thierry Coulbois, Arnaud Hilion To cite this version: Thierry Coulbois, Arnaud Hilion. Ergodic currents dual to a real tree. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2016, 36 (3), pp.745-766. 10.1017/etds.2014.78. hal- 01481866 HAL Id: hal-01481866 https://hal.archives-ouvertes.fr/hal-01481866 Submitted on 3 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ERGODIC CURRENTS DUAL TO A REAL TREE THIERRY COULBOIS, ARNAUD HILION Abstract. Let T be an R-tree with dense orbits in the boundary of Outer space. When the free group FN acts freely on T , we prove that the number of projective classes of ergodic currents dual to T is bounded above by 3N − 5. We combine Rips induction and splitting induction to define unfolding induction for such an R-tree T . Given a current µ dual to T , the unfolding induction produces a sequence of approximations converging towards µ. We also give a unique ergodicity criterion. 1. Introduction 1.1. Main results. Let FN be the free group with N generators. -
Notes on Sela's Work: Limit Groups And
Notes on Sela's work: Limit groups and Makanin-Razborov diagrams Mladen Bestvina∗ Mark Feighn∗ October 31, 2003 Abstract This is the first in a series of papers giving an alternate approach to Zlil Sela's work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G; F) where G is a finitely generated group and F is a non-abelian free group. Contents 1 The Main Theorem 2 2 The Main Proposition 9 3 Review: Measured laminations and R-trees 10 4 Proof of the Main Proposition 18 5 Review: JSJ-theory 19 6 Limit groups are CLG's 22 7 A more geometric approach 23 ∗The authors gratefully acknowledge support of the National Science Foundation. Pre- liminary Version. 1 1 The Main Theorem This is the first of a series of papers giving an alternative approach to Zlil Sela's work on the Tarski problems [31, 30, 32, 24, 25, 26, 27, 28]. The present paper is an exposition of the following result of Kharlampovich-Myasnikov [9, 10] and Sela [30]: Theorem. Let G be a finitely generated non-free group. There is a finite collection fqi : G ! Γig of proper quotients of G such that, for any homo- morphism f from G to a free group F , there is α 2 Aut(G) such that fα factors through some qi. A more precise statement is given in the Main Theorem. Our approach, though similar to Sela's, differs in several aspects: notably a different measure of complexity and a more geometric proof which avoids the use of the full Rips theory for finitely generated groups acting on R-trees, see Section 7. -
Homology Homomorphism
|| Computations Induced (Co)homology homomorphism Daher Al Baydli Department of Mathematics National University of Ireland,Galway Feberuary, 2017 Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 1 / 11 Overview Resolution Chain and cochain complex homology cohomology Induce (Co)homology homomorphism Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 2 / 11 Resolution Definition Let G be a group and Z be the group of integers considered as a trivial ZG-module. The map : ZG −! Z from the integral group ring to Z, given by Σmg g 7−! Σmg , is called the augmentation. Definition Let G be a group. A free ZG-resolution of Z is an exact sequence of free G G @n+1 G @n G @n−1 @2 G @1 ZG-modules R∗ : · · · −! Rn+1 −! Rn −! Rn−1 −! · · · −! R1 −! G G R0 −! R−1 = Z −! 0 G with each Ri a free ZG-module for all i ≥ 0. Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 3 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. For each n 2 Z, the nth homology module of C is defined to be the quotient module Ker@n Hn(C) = Im@n+1 G We can also construct an induced cochain complex HomZG (R∗ ; A) of abelian groups: G G δn G δn−1 HomZG (R∗ ; A): · · · − HomZG (Rn+1; A) − HomZG (Rn ; A) − G δn−2 δ1 G δ0 G HomZG (Rn−1; A) − · · · − HomZG (R1 ; A) − HomZG (R0 ; A) Chain and cochain complex G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G @n+1 G @n abelian groups: R∗ ⊗ZG A : · · · −! Rn+1 ⊗ZG A −! Rn ⊗ZG A −! G @n−1 @2 G @1 G Rn−1 ⊗ZG A −! · · · −! R1 ⊗ZG A −! R0 ⊗ZG A Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. -
THE DIMENSION of the TORELLI GROUP 1. Introduction Let S Be A
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 1, January 2010, Pages 61–105 S 0894-0347(09)00643-2 Article electronically published on July 10, 2009 THE DIMENSION OF THE TORELLI GROUP MLADEN BESTVINA, KAI-UWE BUX, AND DAN MARGALIT 1. Introduction Let S be a surface (unless specified otherwise, we take all surfaces to be con- nected, orientable, and of finite type). Let Mod(S)bethemapping class group of S, + + defined as π0(Homeo (S)), where Homeo (S) is the group of orientation-preserving homeomorphisms of S.TheTorelli group I(S) is the kernel of the natural action of Mod(S)onH1(S, Z). When S is Sg, the closed surface of genus g, this action is symplectic (it preserves the algebraic intersection number), and it is a classical fact that Mod(Sg) surjects onto Sp(2g, Z). All of this information is encoded in the following short exact sequence: 1 −→I(Sg) −→ Mod(Sg) −→ Sp(2g, Z) −→ 1. Cohomological dimension. For a group G,wedenotebycd(G)itscohomological dimension, which is the supremum over all n so that there exists a G-module M with Hn(G, M) =0. Theorem A. For g ≥ 2, we have cd(I(Sg)) = 3g − 5. ∼ Since Mod(S0)=1,wehaveI(S0) = 1. Also, it is a classical fact that Mod(S1) = Sp(2, Z)=SL(2, Z), and so I(S1)isalsotrivial. The lower bound for Theorem A was already given by Mess [36] in an unpublished paper from 1990. Theorem 1.1 (Mess). For g ≥ 2, we have cd(I(Sg)) ≥ 3g − 5.