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Geometric Methods in Heegaard Theory
GEOMETRIC METHODS IN HEEGAARD THEORY TOBIAS HOLCK COLDING, DAVID GABAI, AND DANIEL KETOVER Abstract. We survey some recent geometric methods for studying Heegaard splittings of 3-manifolds 0. Introduction A Heegaard splitting of a closed orientable 3-manifold M is a decomposition of M into two handlebodies H0 and H1 which intersect exactly along their boundaries. This way of thinking about 3-manifolds (though in a slightly different form) was discovered by Poul Heegaard in his forward looking 1898 Ph. D. thesis [Hee1]. He wanted to classify 3-manifolds via their diagrams. In his words1: Vi vende tilbage til Diagrammet. Den Opgave, der burde løses, var at reducere det til en Normalform; det er ikke lykkedes mig at finde en saadan, men jeg skal dog fremsætte nogle Bemærkninger angaaende Opgavens Løsning. \We will return to the diagram. The problem that ought to be solved was to reduce it to the normal form; I have not succeeded in finding such a way but I shall express some remarks about the problem's solution." For more details and a historical overview see [Go], and [Zie]. See also the French translation [Hee2], the English translation [Mun] and the partial English translation [Prz]. See also the encyclopedia article of Dehn and Heegaard [DH]. In his 1932 ICM address in Zurich [Ax], J. W. Alexander asked to determine in how many essentially different ways a canonical region can be traced in a manifold or in modern language how many different isotopy classes are there for splittings of a given genus. He viewed this as a step towards Heegaard's program. -
Homotopy Type of the Complex of Free Factors of a Free Group
Research Collection Other Journal Item Homotopy type of the complex of free factors of a free group Author(s): Gupta, Radhika; Brück, Benjamin Publication Date: 2020-12 Permanent Link: https://doi.org/10.3929/ethz-b-000463712 Originally published in: Proceedings of the London Mathematical Society 121(6), http://doi.org/10.1112/plms.12381 Rights / License: Creative Commons Attribution 4.0 International This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Proc. London Math. Soc. (3) 121 (2020) 1737–1765 doi:10.1112/plms.12381 Homotopy type of the complex of free factors of a free group Benjamin Br¨uck and Radhika Gupta Abstract We show that the complex of free factors of a free group of rank n 2 is homotopy equivalent to a wedge of spheres of dimension n − 2. We also prove that for n 2, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n − 2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible. 1. Introduction Let F be the free group of finite rank n. A free factor of F is a subgroup A such that F = A ∗ B for some subgroup B of F. Let [.] denote the conjugacy class of a subgroup of F. Define Fn to be the partially ordered set (poset) of conjugacy classes of proper, non-trivial free factors of F where [A] [B] if for suitable representatives, one has A ⊆ B. -
On the Topology of Ending Lamination Space
1 ON THE TOPOLOGY OF ENDING LAMINATION SPACE DAVID GABAI Abstract. We show that if S is a finite type orientable surface of genus g and p punctures where 3g + p ≥ 5, then EL(S) is (n − 1)-connected and (n − 1)- locally connected, where dim(PML(S)) = 2n + 1 = 6g + 2p − 7. Furthermore, if g = 0, then EL(S) is homeomorphic to the p−4 dimensional Nobeling space. 0. Introduction This paper is about the topology of the space EL(S) of ending laminations on a finite type hyperbolic surface, i.e. a complete hyperbolic surface S of genus-g with p punctures. An ending lamination is a geodesic lamination L in S that is minimal and filling, i.e. every leaf of L is dense in L and any simple closed geodesic in S nontrivally intersects L transversely. Since Thurston's seminal work on surface automorphisms in the mid 1970's, laminations in surfaces have played central roles in low dimensional topology, hy- perbolic geometry, geometric group theory and the theory of mapping class groups. From many points of view, the ending laminations are the most interesting lam- inations. For example, the stable and unstable laminations of a pseudo Anosov mapping class are ending laminations [Th1] and associated to a degenerate end of a complete hyperbolic 3-manifold with finitely generated fundamental group is an ending lamination [Th4], [Bon]. Also, every ending lamination arises in this manner [BCM]. The Hausdorff metric on closed sets induces a metric topology on EL(S). Here, two elements L1, L2 in EL(S) are close if each point in L1 is close to a point of L2 and vice versa. -
On Hyperbolicity of Free Splitting and Free Factor Complexes
ON HYPERBOLICITY OF FREE SPLITTING AND FREE FACTOR COMPLEXES ILYA KAPOVICH AND KASRA RAFI Abstract. We show how to derive hyperbolicity of the free factor complex of FN from the Handel-Mosher proof of hyperbolicity of the free splitting complex of FN , thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that the natural map from the free splitting complex to free factor complex sends geodesics to quasi-geodesics. 1. Introduction The notion of a curve complex, introduced by Harvey [9] in late 1970s, plays a key role in the study of hyperbolic surfaces, mapping class group and the Teichm¨ullerspace. If S is a compact connected oriented surface, the curve complex C(S) of S is a simplicial complex whose vertices are isotopy classes of essential non-peripheral simple closed curves. A collection [α0];:::; [αn] of (n + 1) distinct vertices of C(S) spans an n{simplex in C(S) if there exist representatives α0; : : : ; αn of these isotopy classes such that for all i 6= j the curves αi and αj are disjoint. (The definition of C(S) is a little different for several surfaces of small genus). The complex C(S) is finite-dimensional but not locally finite, and it comes equipped with a natural action of the mapping class group Mod(S) by simplicial automorphisms. It turns out that the geometry of C(S) is closely related to the geometry of the Teichm¨ullerspace T (S) and also of the mapping class group itself. The curve complex is a basic tool in modern Teichmuller theory, and has also found numerous applications in the study of 3-manifolds and of Kleinian groups. -
3-Manifold Groups
3-Manifold Groups Matthias Aschenbrenner Stefan Friedl Henry Wilton University of California, Los Angeles, California, USA E-mail address: [email protected] Fakultat¨ fur¨ Mathematik, Universitat¨ Regensburg, Germany E-mail address: [email protected] Department of Pure Mathematics and Mathematical Statistics, Cam- bridge University, United Kingdom E-mail address: [email protected] Abstract. We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise. Contents Introduction 1 Chapter 1. Decomposition Theorems 7 1.1. Topological and smooth 3-manifolds 7 1.2. The Prime Decomposition Theorem 8 1.3. The Loop Theorem and the Sphere Theorem 9 1.4. Preliminary observations about 3-manifold groups 10 1.5. Seifert fibered manifolds 11 1.6. The JSJ-Decomposition Theorem 14 1.7. The Geometrization Theorem 16 1.8. Geometric 3-manifolds 20 1.9. The Geometric Decomposition Theorem 21 1.10. The Geometrization Theorem for fibered 3-manifolds 24 1.11. 3-manifolds with (virtually) solvable fundamental group 26 Chapter 2. The Classification of 3-Manifolds by their Fundamental Groups 29 2.1. Closed 3-manifolds and fundamental groups 29 2.2. Peripheral structures and 3-manifolds with boundary 31 2.3. Submanifolds and subgroups 32 2.4. Properties of 3-manifolds and their fundamental groups 32 2.5. Centralizers 35 Chapter 3. 3-manifold groups after Geometrization 41 3.1. Definitions and conventions 42 3.2. Justifications 45 3.3. Additional results and implications 59 Chapter 4. The Work of Agol, Kahn{Markovic, and Wise 63 4.1. -
January 2013 Prizes and Awards
January 2013 Prizes and Awards 4:25 P.M., Thursday, January 10, 2013 PROGRAM SUMMARY OF AWARDS OPENING REMARKS FOR AMS Eric Friedlander, President LEVI L. CONANT PRIZE: JOHN BAEZ, JOHN HUERTA American Mathematical Society E. H. MOORE RESEARCH ARTICLE PRIZE: MICHAEL LARSEN, RICHARD PINK DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY DAVID P. ROBBINS PRIZE: ALEXANDER RAZBOROV TEACHING OF MATHEMATICS RUTH LYTTLE SATTER PRIZE IN MATHEMATICS: MARYAM MIRZAKHANI Mathematical Association of America LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT: YAKOV SINAI EULER BOOK PRIZE LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION: JOHN GUCKENHEIMER, PHILIP HOLMES Mathematical Association of America LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH: SAHARON SHELAH LEVI L. CONANT PRIZE OSWALD VEBLEN PRIZE IN GEOMETRY: IAN AGOL, DANIEL WISE American Mathematical Society DAVID P. ROBBINS PRIZE FOR AMS-SIAM American Mathematical Society NORBERT WIENER PRIZE IN APPLIED MATHEMATICS: ANDREW J. MAJDA OSWALD VEBLEN PRIZE IN GEOMETRY FOR AMS-MAA-SIAM American Mathematical Society FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY ALICE T. SCHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN AN UNDERGRADUATE STUDENT: FAN WEI Association for Women in Mathematics FOR AWM LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION: AMY COHEN Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE -
Submanifolds of Euclidean Space with Parallel Second Fundamental Form
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 1, March 1972 SUBMANIFOLDS OF EUCLIDEAN SPACE WITH PARALLEL SECOND FUNDAMENTAL FORM JAAK VILMS1 In this paper necessary conditions are given for a complete Rie- mannian manifold M" to admit an isometric immersion into R*+r with parallel second fundamental form. Namely, it is shown that M" must be affinely equivalent either to a totally geodesic sub- manifold of the Grassmann manifold G(n,p), or to a fibre bundle over such a submanifold, with Euclidean space as fibre and the structure being close to a product. (An affine equivalence is a dif- feomorphism that preserves Riemannian connections.) The proof depends on the auxiliary result that the second fundamental form is parallel iff the Gauss map is a totally geodesic map. 1. The main result. Let i :Mn-+Rn+p be an immersion of a manifold Mn into Euclidean space, and endow M with the induced Riemannian metric. The second fundamental form ¡x of the immersion ; is a symmetric section of the bundle of bilinear maps L2(TM, TM; A/M), where TM and NM denote the tangent and normal bundles of M, respectively [4]. We iden- tify this bundle of maps with the isomorphic bundle L(TM, L(TM, NM)), and <xwith the corresponding section. At each point of M, the kernel of x (in the second interpretation) is called the relative nullity space, and its dimension k, the relative nullity index [2]. We say the second fundamental form a is parallel if 7>oc=0 (D always denotes the appropriate covariant derivative). -
Actions of Mapping Class Groups Athanase Papadopoulos
Actions of mapping class groups Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Actions of mapping class groups. L. Ji, A. Papadopoulos and S.-T. Yau. Handbook of Group Actions, Vol. I, 31, Higher Education Press; International Press, p. 189-248., 2014, Advanced Lectures in Mathematics, 978-7-04-041363-2. hal-01027411 HAL Id: hal-01027411 https://hal.archives-ouvertes.fr/hal-01027411 Submitted on 21 Jul 2014 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. ACTIONS OF MAPPING CLASS GROUPS ATHANASE PAPADOPOULOS Abstract. This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston’s sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups. -
1. the Ending Lamination Conjecture 1 2
THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II: THE ENDING LAMINATION CONJECTURE JEFFREY F. BROCK, RICHARD D. CANARY, AND YAIR N. MINSKY Abstract. Thurston's Ending Lamination Conjecture states that a hy- perbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups. The main ingredi- ent is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in [47], and a sub- sequent paper [15] will establish the Ending Lamination Conjecture in general. Contents 1. The ending lamination conjecture 1 2. Background and statements 8 3. Knotting and partial order of subsurfaces 25 4. Cut systems and partial orders 49 5. Regions and addresses 66 6. Uniform embeddings of Lipschitz surfaces 76 7. Insulating regions 91 8. Proof of the bilipschitz model theorem 99 9. Proofs of the main theorems 118 10. Corollaries 119 References 123 1. The ending lamination conjecture In the late 1970's Thurston formulated a conjectural classification scheme for all hyperbolic 3-manifolds with finitely generated fundamental group. The picture proposed by Thurston generalized what had hitherto been un- derstood, through the work of Ahlfors [3], Bers [10], Kra [34], Marden [37], Maskit [38], Mostow [51], Prasad [55], Thurston [67] and others, about geo- metrically finite hyperbolic 3-manifolds. Thurston's scheme proposes end invariants which encode the asymptotic geometry of the ends of the manifold, and which generalize the Riemann Date: December 7, 2004. Partially supported by NSF grants DMS-0354288, DMS-0203698 and DMS-0203976. -
Marden's Tameness Conjecture
MARDEN'S TAMENESS CONJECTURE: HISTORY AND APPLICATIONS RICHARD D. CANARY Abstract. Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We will survey the history of work on this conjecture and discuss its many appli- cations. 1. Introduction In a seminal paper, published in the Annals of Mathematics in 1974, Al Marden [65] conjectured that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. This conjecture evolved into one of the central conjectures in the theory of hyperbolic 3-manifolds. For example, Mar- den's Tameness Conjecture implies Ahlfors' Measure Conjecture (which we will discuss later). It is a crucial piece in the recently completed classification of hyperbolic 3-manifolds with finitely generated funda- mental group. It also has important applications to geometry and dy- namics of hyperbolic 3-manifolds and gives important group-theoretic information about fundamental groups of hyperbolic 3-manifolds. There is a long history of partial results in the direction of Marden's Tameness Conjecture and it was recently completely established by Agol [1] and Calegari-Gabai [27]. In this brief expository paper, we will survey the history of these results and discuss some of the most important applications. Outline of paper: In section 2, we recall basic definitions from the theory of hyperbolic 3-manifolds. In section 3, we construct a 3- manifold with finitely generated fundamental group which is not home- omorphic to the interior of a compact 3-manifold. -
Marden's Tameness Conjecture: History and Applications
MARDEN’S TAMENESS CONJECTURE: HISTORY AND APPLICATIONS RICHARD D. CANARY Abstract. Marden’s Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We will survey the history of work on this conjecture and discuss its many appli- cations. 1. Introduction In a seminal paper, published in the Annals of Mathematics in 1974, Al Marden [65] conjectured that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. This conjecture evolved into one of the central conjectures in the theory of hyperbolic 3-manifolds. For example, Mar- den’s Tameness Conjecture implies Ahlfors’ Measure Conjecture (which we will discuss later). It is a crucial piece in the recently completed classification of hyperbolic 3-manifolds with finitely generated funda- mental group. It also has important applications to geometry and dy- namics of hyperbolic 3-manifolds and gives important group-theoretic information about fundamental groups of hyperbolic 3-manifolds. There is a long history of partial results in the direction of Marden’s Tameness Conjecture and it was recently completely established by arXiv:1008.0118v1 [math.GT] 31 Jul 2010 Agol [1] and Calegari-Gabai [27]. In this brief expository paper, we will survey the history of these results and discuss some of the most important applications. Outline of paper: In section 2, we recall basic definitions from the theory of hyperbolic 3-manifolds. In section 3, we construct a 3- manifold with finitely generated fundamental group which is not home- omorphic to the interior of a compact 3-manifold. -
Book of Abstracts
Georgian National Georgian Mathematical Ivane Javakhishvili Academy of Sciences Union Tbilisi State University Caucasian Mathematics Conference CMC I BOOK OF ABSTRACTS Tbilisi, September 5 – 6 2014 Organizers: European Mathematical Society Armenian Mathematical Union Azerbaijan Mathematical Union Georgian Mathematical Union Iranian Mathematical Society Moscow Mathematical Society Turkish Mathematical Society Sponsors: Georgian National Academy of Sciences Ivane Javakhishvili Tbilisi State University Steering Committee: Carles Casacuberta (ex officio; Chair of the EMS Committee for European Solidarity), Mohammed Ali Dehghan (President of Iranian Mathematical Society), Roland Duduchava (President of the Georgian Mathematical Union), Tigran Harutyunyan (President of the Armenian Mathematical Union), Misir Jamayil oglu Mardanov (Representative of the Azerbaijan Mathematical Union), Marta Sanz-Sole (ex officio; President of the European Mathematical Society), Armen Sergeev (Representative of the Moscow Mathematical Society and EMS), Betül Tanbay (President of the Turkish Mathematical Society). Local Organizing Committee: Tengiz Buchukuri (IT Responsible), Tinatin Davitashvili (Scientific Secretary), Roland Duduchava (Chairman), David Natroshvili (Vice chairman), Levan Sigua (Treasurer). Editors: Guram Gogishvili, Vazha Tarieladze, Maia Japoshvili Cover Design: David Sulakvelidze Contents Abstracts of Invited Talks 13 Maria J. Esteban, Symmetry and Symmetry Breaking for Optimizers of Functional Inequalities . 15 Mohammad Sal Moslehian, Recent Developments in Operator Inequalities . 15 Garib N. Murshudov, Some Application of Mathematics to Structural Biology 16 Dmitri Orlov, Categories in Geometry and Physics: Mirror Symmetry, D- Branes and Landau–Ginzburg Models . 17 Samson Shatashvili, Supersymmetric Gauge Theories and the Quantisation of Integrable Systems . 18 Leon A. Takhtajan, On Bott–Chern and Chern–Simons Characteristic Forms 19 Cem Yalçin Yildirim, Small Gaps Between Primes: the GPY Method and Recent Advancements Over it .