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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. -
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. -
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 of Hom Complexes
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 10 Homology of Hom Complexes Mychael Sanchez New Mexico State University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Sanchez, Mychael (2012) "Homology of Hom Complexes," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 10. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/10 Rose- Hulman Undergraduate Mathematics Journal Homology of Hom Complexes Mychael Sancheza Volume 13, No. 1, Spring 2012 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal New Mexico State University Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Homology of Hom Complexes Mychael Sanchez Abstract. The hom complex Hom (G; K) is the order complex of the poset com- posed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups. Acknowledgements: I acknowledge my advisor, Dr. Dan Ramras and the National Science Foundation for funding this research through Dr. Ramras' grants, DMS-1057557 and DMS-0968766. Page 212 RHIT Undergrad. Math. J., Vol. 13, No. 1 1 Introduction In 1978, László Lovász proved the Kneser Conjecture [8]. His proof involves associating to a graph G its neighborhood complex N (G) and using its topological properties to locate obstructions to graph colorings. This proof illustrates the fruitful interaction between graph theory, combinatorics and topology. -
THE SYMMETRIES of OUTER SPACE Martin R. Bridson* and Karen Vogtmann**
THE SYMMETRIES OF OUTER SPACE Martin R. Bridson* and Karen Vogtmann** ABSTRACT. For n 3, the natural map Out(Fn) → Aut(Kn) from the outer automorphism group of the free group of rank n to the group of simplicial auto- morphisms of the spine of outer space is an isomorphism. §1. Introduction If a eld F has no non-trivial automorphisms, then the fundamental theorem of projective geometry states that the group of incidence-preserving bijections of the projective space of dimension n over F is precisely PGL(n, F ). In the early nineteen seventies Jacques Tits proved a far-reaching generalization of this theorem: under suitable hypotheses, the full group of simplicial automorphisms of the spherical building associated to an algebraic group is equal to the algebraic group — see [T, p.VIII]. Tits’s theorem implies strong rigidity results for lattices in higher-rank — see [M]. There is a well-developed analogy between arithmetic groups on the one hand and mapping class groups and (outer) automorphism groups of free groups on the other. In this analogy, the role played by the symmetric space in the classical setting is played by the Teichmuller space in the case of case of mapping class groups and by Culler and Vogtmann’s outer space in the case of Out(Fn). Royden’s Theorem (see [R] and [EK]) states that the full isometry group of the Teichmuller space associated to a compact surface of genus at least two (with the Teichmuller metric) is the mapping class group of the surface. An elegant proof of Royden’s theorem was given recently by N. -
Constructing Covers of the Rose
Constructing Covers of the rose. Robert Kropholler April 25, 2017 This was discussed in class but does not appear in any of the lecture notes. I will outline the constructions and proof here. Definition 0.1. Let F (X) be the free group on the set X. Assume jXj = n. Let H be any subgroup of F (X), we define the Schrier graph G(H) as follows: • The vertices of G(H) are the left cosets of H in G. • Two cosets gH; g0H are joined by an edge if there exists an x 2 X such that xgH = g0H. It should be noted that the Schrier graph of a normal subgroup N is the Cayley graph of G=N with respect to the generating set X. Let Rn be a cell complex with 1 vertex and n edges. We identify the fundamental group of Rn with F (X). Pick a bijection between the edges of X and give each edge an orientation. Proposition 0.2. The graph G(H) is a cover of Rn. Proof. We can label each edge with an element of X and give it an orientation so that the edge (gH; xgH) points towards xgH. We can then define a map p : G(H) ! Rn. This sends each vertex of G(H) to the unique vertex of Rn and each edge to the edge with same label by an orientation preserving homeomorphism. This defines a covering map. We must check that every point has a neighbourhood such that p is a homeomorphism on this neighbourhood. -
The Cohomology of Automorphism Groups of Free Groups
The cohomology of automorphism groups of free groups Karen Vogtmann∗ Abstract. There are intriguing analogies between automorphism groups of finitely gen- erated free groups and mapping class groups of surfaces on the one hand, and arithmetic groups such as GL(n, Z) on the other. We explore aspects of these analogies, focusing on cohomological properties. Each cohomological feature is studied with the aid of topolog- ical and geometric constructions closely related to the groups. These constructions often reveal unexpected connections with other areas of mathematics. Mathematics Subject Classification (2000). Primary 20F65; Secondary, 20F28. Keywords. Automorphism groups of free groups, Outer space, group cohomology. 1. Introduction In the 1920s and 30s Jakob Nielsen, J. H. C. Whitehead and Wilhelm Magnus in- vented many beautiful combinatorial and topological techniques in their efforts to understand groups of automorphisms of finitely-generated free groups, a tradition which was supplemented by new ideas of J. Stallings in the 1970s and early 1980s. Over the last 20 years mathematicians have been combining these ideas with others motivated by both the theory of arithmetic groups and that of surface mapping class groups. The result has been a surge of activity which has greatly expanded our understanding of these groups and of their relation to many areas of mathe- matics, from number theory to homotopy theory, Lie algebras to bio-mathematics, mathematical physics to low-dimensional topology and geometric group theory. In this article I will focus on progress which has been made in determining cohomological properties of automorphism groups of free groups, and try to in- dicate how this work is connected to some of the areas mentioned above. -
Detecting Fully Irreducible Automorphisms: a Polynomial Time Algorithm
DETECTING FULLY IRREDUCIBLE AUTOMORPHISMS: A POLYNOMIAL TIME ALGORITHM. WITH AN APPENDIX BY MARK C. BELL. ILYA KAPOVICH Abstract. In [30] we produced an algorithm for deciding whether or not an element ' 2 Out(FN ) is an iwip (\fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this paper we refine the algorithm from [30] by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed N ≥ 3, an algorithm which, given a topological representative f of an element ' of Out(FN ), decides in polynomial time in terms of the \size" of f, whether or not ' is fully irreducible. In addition, we provide a train track criterion of being fully irreducible which covers all fully irreducible elements of Out(FN ), including both atoroidal and non-atoroidal ones. We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train track map, and estimate the complexity of this algorithm. An appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina-Handel algorithm[3] for finding either an irreducible train track representative or a topological reduction. 1. Introduction For an integer N ≥ 2 an outer automorphism ' 2 Out(FN ) is called fully irreducible if there no positive power of ' preserves the conjugacy class of any proper free factor of FN . -
Moduli Spaces of Colored Graphs
MODULI SPACES OF COLORED GRAPHS MARKO BERGHOFF AND MAX MUHLBAUER¨ Abstract. We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by var- ious conditions on the coloring of their graphs. The motivation for this is to study Feynman integrals in quantum field theory using the combinatorial structure of these moduli spaces. Here a family of graphs is specified by the allowed Feynman diagrams in a particular quantum field theory such as (mas- sive) scalar fields or quantum electrodynamics. The resulting spaces are cell complexes with a rich and interesting combinatorial structure. We treat some examples in detail and discuss their topological properties, connectivity and homology groups. 1. Introduction The purpose of this article is to define and study moduli spaces of colored graphs in the spirit of [HV98] and [CHKV16] where such spaces for uncolored graphs were used to study the homology of automorphism groups of free groups. Our motivation stems from the connection between these constructions in geometric group theory and the study of Feynman integrals as pointed out in [BK15]. Let us begin with a brief sketch of these two fields and what is known so far about their relation. 1.1. Background. The analysis of Feynman integrals as complex functions of their external parameters (e.g. momenta and masses) is a special instance of a very gen- eral problem, understanding the analytic structure of functions defined by integrals. That is, given complex manifolds X; T and a function f : T ! C defined by Z f(t) = !t; Γ what can be deduced about f from studying the integration contour Γ ⊂ X and the integrand !t 2 Ω(X)? There is a mathematical treatment for a class of well-behaved cases [Pha11], arXiv:1809.09954v3 [math.AT] 2 Jul 2019 but unfortunately Feynman integrals are generally too complicated to answer this question thoroughly. -
Knot Theory and the Alexander Polynomial
Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ............................... -
Perspectives in Lie Theory Program of Activities
INdAM Intensive research period Perspectives in Lie Theory Program of activities Session 3: Algebraic topology, geometric and combinatorial group theory Period: February 8 { February 28, 2015 All talks will be held in Aula Dini, Palazzo del Castelletto. Monday, February 9, 2015 • 10:00- 10:40, registration • 10:40, coffee break. • 11:10- 12:50, Vic Reiner, Reflection groups and finite general linear groups, lecture 1. • 15:00- 16:00, Michael Falk, Rigidity of arrangement complements • 16:00, coffee break. • 16:30- 17:30, Max Wakefield, Kazhdan-Lusztig polynomial of a matroid • 17:30- 18:30, Angela Carnevale, Odd length: proof of two conjectures and properties (young semi- nar). • 18:45- 20:30, Welcome drink (Sala del Gran Priore, Palazzo della Carovana, Piazza dei Cavalieri) Tuesday, February 10, 2015 • 9:50-10:40, Ulrike Tillmann, Homology of mapping class groups and diffeomorphism groups, lecture 1. • 10:40, coffee break. • 11:10- 12:50, Karen Vogtmann, On the cohomology of automorphism groups of free groups, lecture 1. • 15:00- 16:00, Tony Bahri,New approaches to the cohomology of polyhedral products • 16:00, coffee break. • 16:30- 17:30, Alexandru Dimca, On the fundamental group of algebraic varieties • 17:30- 18:30, Nancy Abdallah, Cohomology of Algebraic Plane Curves (young seminar). Wednesday, February 11, 2015 • 9:00- 10:40, Vic Reiner, Reflection groups and finite general linear groups, lecture 2. • 10:40, coffee break. • 11:10- 12:50, Ulrike Tillmann, Homology of mapping class groups and diffeomorphism groups, lec- ture 2 • 15:00- 16:00, Karola Meszaros, Realizing subword complexes via triangulations of root polytopes • 16:00, coffee break.