2020 Leroy P. Steele Prizes
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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 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. -
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. -
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. -
The Legacy of Norbert Wiener: a Centennial Symposium
http://dx.doi.org/10.1090/pspum/060 Selected Titles in This Series 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. -
Karen Vogtmann
CURRICULUM VITAE -KAREN VOGTMANN Mathematics Institute Office: C2.05 Zeeman Bldg. Phone: +44 (0) 2476 532739 University of Warwick Email: [email protected] Coventry CV4 7AL PRINCIPAL FIELDS OF INTEREST Geometric group theory, Low-dimensional topology, Cohomology of groups EDUCATION B.A. University of California, Berkeley 1971 Ph.D. University of California, Berkeley 1977 ACADEMIC POSITIONS University of Warwick, Professor, 9/13 to present Cornell University – Goldwin Smith Professor of Mathematics Emeritus, 7/15 to present – Goldwin Smith Professor of Mathematics, 7/11 to 7/15 – Professor, 1/94 to 7/11 – Associate Professor, 7/87 to 12/93 – Assistant Professor, 7/85 to 6/87 – Visiting Assistant Professor, 9/84 to 6/85 Columbia University, Assistant Professor, 7/79 to 6/86 Brandeis University, Visiting Assistant Professor, 9/78 to 12/78 University of Michigan, Visiting Assistant Professor, 9/77 to 6/78 and 1/79 to 6/79 RESEARCH AND SABBATICAL POSITIONS MSRI, Berkeley, CA 8/19 to 11/19 Newton Institute, Cambridge, Mass, 3/17 to 5/17 MSRI, Berkeley, CA 8/16 to 12/16 Research Professor, ICERM, Providence, RI, 9/13 to 12/13 Freie Universitat¨ Berlin, Berlin, Germany, 6/12 Mittag-Leffler Institute, Stockholm, Sweden, 3/12 to 5/12 Visiting Researcher, Oxford University, Oxford, England, 2/12 Professeur invite, Marseilles, France, 5/11 Hausdorff Institute for Mathematics, 9/09 to 12/09 and 5/10-8/10 Mathematical Sciences Research Institute, Berkeley, CA, 8/07-12/07 I.H.E.S., Bures-sur-Yvette, France 3/04 Professeur Invite,´ Marseilles, France, 3/00 Mathematical Sciences Research Institute, Berkeley, 1/95 to 7/95 I.H.E.S., Bures-sur-Yvette, France, 1/93-8/93 Chercheur, C.N.R.S., E.N.S. -
Prizes and Awards
DENVER • JAN 15–18, 2020 January 2020 Prizes and Awards 4:25 PM, Thursday, January 16, 2020 PROGRAM OPENING REMARKS Michael Dorff, Mathematical Association of America AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society BÔCHER MEMORIAL PRIZE American Mathematical Society CHEVALLEY PRIZE IN LIE THEORY American Mathematical Society FRANK NELSON COLE PRIZE IN NUMBER THEORY American Mathematical Society LEONARD EISENBUD PRIZE FOR MATHEMATICS AND PHYSICS American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society JOSEPH L. DOOB PRIZE American Mathematical Society LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society LOUISE HAY AWARD FOR CONTRIBUTION TO MATHEMATICS EDUCATION Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS Association for Women in Mathematics MICROSOFT RESEARCH PRIZE IN ALGEBRA AND NUMBER THEORY Association for Women in Mathematics SADOSKY RESEARCH PRIZE IN ANALYSIS Association for Women in Mathematics FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics COMMUNICATIONS AWARD Joint Policy Board for Mathematics CHAUVENET PRIZE Mathematical Association of America DAVID P. R OBBINS PRIZE Mathematical Association of America EULER BOOK PRIZE Mathematical Association of America DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America YUEH-GIN GUNG AND DR.CHARLES Y. HU AWARD FOR DISTINGUISHED SERVICE TO MATHEMATICS Mathematical Association of America CLOSING REMARKS Jill C. -
Combinatorial and Geometric Group Theory
Combinatorial and Geometric Group Theory Vanderbilt University Nashville, TN, USA May 5–10, 2006 Contents V. A. Artamonov . 1 Goulnara N. Arzhantseva . 1 Varujan Atabekian . 2 Yuri Bahturin . 2 Angela Barnhill . 2 Gilbert Baumslag . 3 Jason Behrstock . 3 Igor Belegradek . 3 Collin Bleak . 4 Alexander Borisov . 4 Lewis Bowen . 5 Nikolay Brodskiy . 5 Kai-Uwe Bux . 5 Ruth Charney . 6 Yves de Cornulier . 7 Maciej Czarnecki . 7 Peter John Davidson . 7 Karel Dekimpe . 8 Galina Deryabina . 8 Volker Diekert . 9 Alexander Dranishnikov . 9 Mikhail Ershov . 9 Daniel Farley . 10 Alexander Fel’shtyn . 10 Stefan Forcey . 11 Max Forester . 11 Koji Fujiwara . 12 Rostislav Grigorchuk . 12 Victor Guba . 12 Dan Guralnik . 13 Jose Higes . 13 Sergei Ivanov . 14 Arye Juhasz . 14 Michael Kapovich . 14 Ilya Kazachkov . 15 i Olga Kharlampovich . 15 Anton Klyachko . 15 Alexei Krasilnikov . 16 Leonid Kurdachenko . 16 Yuri Kuzmin . 17 Namhee Kwon . 17 Yuriy Leonov . 18 Rena Levitt . 19 Artem Lopatin . 19 Alex Lubotzky . 19 Alex Lubotzky . 20 Olga Macedonska . 20 Sergey Maksymenko . 20 Keivan Mallahi-Karai . 21 Jason Manning . 21 Luda Markus-Epstein . 21 John Meakin . 22 Alexei Miasnikov . 22 Michael Mihalik . 22 Vahagn H. Mikaelian . 23 Ashot Minasyan . 23 Igor Mineyev . 24 Atish Mitra . 24 Nicolas Monod . 24 Alexey Muranov . 25 Bernhard M¨uhlherr . 25 Volodymyr Nekrashevych . 25 Graham Niblo . 26 Alexander Olshanskii . 26 Denis Osin . 27 Panos Papasoglu . 27 Alexandra Pettet . 27 Boris Plotkin . 28 Eugene Plotkin . 28 John Ratcliffe . 29 Vladimir Remeslennikov . 29 Tim Riley . 29 Nikolay Romanovskiy . 30 Lucas Sabalka . 30 Mark Sapir . 31 Paul E. Schupp . 31 Denis Serbin . 32 Lev Shneerson . -
January 2007 Prizes and Awards
January 2007 Prizes and Awards 4:25 P.M., Saturday, January 6, 2007 MATHEMATICAL ASSOCIATION OF AMERICA DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS In 1991, the Mathematical Association of America instituted the Deborah and Franklin Tepper Haimo Awards for Distinguished College or University Teaching of Mathematics in order to honor college or university teachers who have been widely recognized as extraordinarily successful and whose teaching effectiveness has been shown to have had influence beyond their own institutions. Deborah Tepper Haimo was president of the Association, 1991–1992. Citation Jennifer Quinn Jennifer Quinn has a contagious enthusiasm that draws students to mathematics. The joy she takes in all things mathematical is reflected in her classes, her presentations, her publications, her videos and her on-line materials. Her class assignments often include nonstandard activities, such as creating time line entries for historic math events, or acting out scenes from the book Proofs and Refutations. One student created a children’s story about prime numbers and another produced a video documentary about students’ perceptions of math. A student who had her for six classes says, “I hope to become a teacher after finishing my master’s degree and I would be thrilled if I were able to come anywhere close to being as great a teacher as she is.” Jenny developed a variety of courses at Occidental College. Working with members of the physics department and funded by an NSF grant, she helped develop a combined yearlong course in calculus and mechanics. She also developed a course on “Mathematics as a Liberal Art” which included computer discussions, writing assignments, and other means to draw technophobes into the course. -
Colloquium Karen Vogtmann, Cornell Maps Into and out of Automorphism Groups of Free Groups
COLLOQUIUM KAREN VOGTMANN, CORNELL MAPS INTO AND OUT OF AUTOMORPHISM GROUPS OF FREE GROUPS GABRIEL C. DRUMMOND-COLE It’s always a pleasure to come to Chicago. I’m happy to be here. I found this picture on your webpage [laughter.] There are strong analogies between outer automorphism groups of free groups and mapping class groups and lattices. I’m going to ignore mapping class groups, although many of the things I’m going to say have to do with those too. Here’s an Escher print of a lattice, but I’m thinking of irreducible lattices in higher rank semisim- ple Lie groups. It’s a discrete subgroup Λ of G so that G/Λ has finite volume. The classical example is Λ = SL(n, Z) ⊂ SL(n, R) which has a finite volume noncompact quotient. Another n n example is Λ = π1(M3) for a hyperbolic manifold. A couple of nonexamples are Z in R , which is reducible, and also not Γ ⊂ P SL(2, R) which are only rank one, so for me n is higher. The other thing is automorphisms of free groups. So for example ρij takes xi to xixj and pre- serves the other generators. There’s also a left multiplication λij which does the same kind of thing, and the permutation eij. Nielsen proved that these three types generate the auto- morphisms. This was in the 1920s and 1930s. Other people interested in this were J. H. C. Whitehead and W. Magnus. They used algebraic and topological methods, but later progress was slow. -
Through Completely Split Train Tracks
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The University of Utah: J. Willard Marriott Digital Library THE GEOMETRY OF Out(FN) THROUGH COMPLETELY SPLIT TRAIN TRACKS by Derrick Wigglesworth A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah May 2018 Copyright c Derrick Wigglesworth 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Derrick Wigglesworth has been approved by the following supervisory committee members: Mladen Bestvina , Chair(s) 21 Feb 2018 Date Approved Kenneth Bromberg , Member 21 Feb 2018 Date Approved Jonathan M. Chaika , Member 21 Feb 2018 Date Approved Mark Feighn , Member 20 Feb 2018 Date Approved Kevin Wortman , Member 21 Feb 2018 Date Approved by Davar Khoshnevisan , Chair/Dean of the Department/College/School of Mathematics and by David B. Kieda , Dean of The Graduate School. ABSTRACT We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for Out(Fn). Then, in joint work with Radhika Gupta, we show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is a direct analog for the cyclic splitting complex of Handel and Mosher’s theorem on loxodromics for the free splitting complex.