DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter http://dx.doi.org/10.1090/surv/029 MATHEMATICAL SURVEYS AND MONOGRAPHS SERIES LIST Volume 1 The problem of moments, 16 Symplectic groups, J. A. Shohat and J. D. Tamarkin O. Timothy O'Meara 2 The theory of rings, 17 Approximation by polynomials N. Jacobson with integral coefficients, 3 Geometry of polynomials, Le Baron O. Ferguson M. Marden 18 Essentials of Brownian motion 4 The theory of valuations, and diffusion, Frank B. Knight O. F. G. Schilling 19 Contributions to the theory of 5 The kernel function and transcendental numbers, Gregory conformal mapping, V. Chudnovsky S. Bergman 20 Partially ordered abelian groups 6 Introduction to the theory of with interpolation, Kenneth R. algebraic functions of one Goodearl variable, C. C. Chevalley 21 The Bieberbach conjecture: 7.1 The algebraic theory of Proceedings of the symposium on semigroups, Volume I, A. H. the occasion of the proof, Albert Clifford and G. B. Preston Baernstein, David Drasin, Peter Duren, and Albert Marden, 7.2 The algebraic theory of Editors semigroups, Volume II, A. H. Clifford and G. B. Preston 22 Noncommutative harmonic analysis, Michael E. Taylor 8 Discontinuous groups and automorphic functions, 23 Introduction to various aspects of J. Lehner degree theory in Banach spaces, E. H. Rothe 9 Linear approximation, Arthur Sard 24 Noetherian rings and their applications, Lance W. Small, 10 An introduction to the analytic Editor theory of numbers, R. Ayoub 25 Asymptotic behavior of dissipative 11 Fixed points and topological systems, Jack K. Hale degree in nonlinear analysis, J. Cronin 26 Operator theory and arithmetic in H°°9 Hari Bercovici 12 Uniform spaces, J. R. Isbell 27 Basic hypergeometric series and 13 Topics in operator theory, applications, Nathan J. Fine A. Brown, R. G. Douglas, C. Pearcy, D. Sarason, A. L. 28 Direct and inverse scattering on Shields; C. Pearcy, Editor the lines, Richard Beals, Percy Deift, and Carlos Tomei 14 Geometric asymptotics, V. Guillemin and S. Sternberg 15 Vector measures, J. Diestel and J. J. Uhl, Jr. Amenability Mathematical Surveys and Monographs Volume 29 Amenability Alan L.T. Paterson American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 43-02, 43A07, 47H10, 22D25, 22D40, 22E15, 22E25, 03E10; Secondary 03E25, 20F05, 20F16, 20F18, 20F19, 20F24, 22D10, 22D15, 22E27, 28C15, 28D05, 43A60, 43A65, 43A80, 46H05, 46H25, 46J10, 46L05, 46L10, 47A35, 53C42, 54D35, 60B05, 60G50, 62F03. Library of Congress Cataloging-in-Publication Data Paterson, Alan L. T., 1944- Amenability. (Mathematical surveys and monographs; no. 29) Includes bibliographies and index. 1. Harmonic analysis. 2. Locally compact groups. I. Title. II. Series. QA403.P37 1988 515'.2433 88-14485 ISBN 0-8218-1529-6 (als. paper) Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. Copyright © 1988 by the American Mathematical Society Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 04 03 02 01 00 Dedicated to MAHLON M. DAY Nicht die Neugierde, nicht die Eitelkeit, nicht die Betrachtung der Nutzlichkeit, nicht die Pflicht und Gewissenhaftigkeit, sondern ein unausloschlicher, ungliicklicher Durst, der sich auj keinen Vergleich enlasst, fiihrt uns zur Wahrheit. G. W. F. Hegel Contents Preface xi List of Further Results xv Plan xvii CHAPTER 0. Introduction: Basic Concepts and Problems of Amenability Theory 1 CHAPTER 1. Amenable Locally Compact Groups and Amenable Semigroups 25 CHAPTER 2. The Algebra of Invariant Means 51 CHAPTER 3. Free Groups and the Amenability of Lie Groups 95 CHAPTER 4. F0lner Conditions 125 CHAPTER 5. Ergodic Theorems for Amenable Locally Compact Groups 195 CHAPTER 6. Locally Compact Groups of Polynomial Growth 217 CHAPTER 7. Sizes of Sets of Invariant Means 269 APPENDICES: A. Nilpotent, Solvable and Semidirect Product Groups 301 B. Lie Groups 305 C. Existence of Borel Cross-Sections 329 D. Mycielski's Theorem 331 E. On the Density of the Exponential Map 333 Some Abbreviations 337 Bibliography 339 Supplementary Bibliography ([S]) 397 Sketched Solutions to Problems 413 Index of Terms 443 Index of Symbols 451 IX Preface The subject of amenability essentially begins with Lebesgue (1904). One of the properties of his integral is a version of the Monotone Convergence Theorem, and Lebesgue asked if this property was really fundamental, that is, if the property follows from the more familiar integral axioms. Now the Monotone Convergence Theorem is equivalent to countable additivity, and so the question is concerned with the existence of a positive, finitely (but not countably) additive, translation invariant measure /i on R with ^([0,1]) = 1. The classical period (1904-1938) is therefore concerned with the study of finitely additive, invariant measure theory. The study of zsorae£r^invariant mea­ sures led to the Banach-Tarski Theorem (1924) and the theory of paradoxical decompositions. The class of amenable groups was introduced and studied by von Neumann (1929) and used to explain why the Banach-Tarski Paradox occurs only for dimension greater than or equal to three. The modern period begins in the 1940s and continues with increasing energy to the present. The main shift is from finitely additive measures to means: integrating a positive, finitely additive measure /i on a set X with pt(X) — 1 gives rise to mean m on X, that is, a continuous linear functional on loo{X) such that ra(l) = 1 = ||m||. The connection between JJL and m is given by n{E) = m(x#), and the correspondence /i —> m is bijective. This shift is of fundamental importance, for it makes available the substantial resources of functional analysis (and eventually of abstract harmonic analysis) to the study of amenability. Von Neumann's definition translates into the language of means and gives the familiar definition of an amenable group: a group G is amenable if and only if there exists a left invariant mean m on G. Here, a mean m is left invariant if m(<j>x) = m(<£), where <f>x{y) — <f>{xy) (y € G) for <f> 6 /oo(G), x G G. The above definition also applies to semigroups, a semigroup admitting a left invariant mean being called left amenable. In the 1940s and 1950s, the subject of amenable groups and amenable semi­ groups was studied by M. M. Day, and his 1957 paper on amenable semigroups is a major landmark. Since that time the subject has developed at a fast pace, as the bibliography will testify. The plan at the beginning of this book will give some idea of the range of mathematics in which the amenability phenomenon has been observed xi xii PREFACE and proved relevant. The amenability phenomenon is protean and is not readily "pigeon-holed". However, the roots of the subject lie in functional analysis and abstract harmonic analysis, and it draws its coherence from these. The ubiquity of amenability ideas and the depth of the mathematics with which the subject is involved seems evidence to the author that here we have a topic of fundamental importance in modern mathematics, one that deserves to be more widely known than it is at present. Until 1984, the only book available on the subject was the influential short account by F. P. Greenleaf [2] (1969). We should also mention Chapter 8 of Reiter's book [R] (1968) and Day's indispensable survey papers [4], [9] of 1957 and 1968 respectively. In addition, §17 of the work on abstract harmonic analysis [HRl] by Hewitt and Ross discusses amenable semigroups and groups. More recently (1984), a book by Jean-Paul Pier on amenable locally compact groups has appeared. As one might expect, there is overlap between Professor Pier's book and the present one. However, the points of view adopted in the two books are very different, as also are much of the contents. Another work that has recently appeared (1985), relevant to amenability, is the elegant book by Stan Wagon dealing with the Banach-Tarski Theorem. The main objectives of the present work are to provide an introduction to the subject as a whole and to go into many of its topics in some depth. While the main area of amenability lies in analysis on locally compact groups, the subject applies to a much wider range of mathematics than that. We have tried to bear this in mind. In particular, applications in the areas of statistics, differential geometry, and operator algebras will be found in Chapters 4, 7, and 1, 2 respectively. Chapter 4 also contains discussions of the two most outstanding theorems in amenability established since 1980—these are in the areas of von Neumann's Conjecture and the Banach-Ruziewicz Problem. We have attempted to describe the main lines of development of the subject, showing what progress or lack of progress has been made in solving the main problems, and raising a number of open problems.
Load more

Recommended publications
  • The Heisenberg Group Fourier Transform
    THE HEISENBERG GROUP FOURIER TRANSFORM NEIL LYALL Contents 1. Fourier transform on Rn 1 2. Fourier analysis on the Heisenberg group 2 2.1. Representations of the Heisenberg group 2 2.2. Group Fourier transform 3 2.3. Convolution and twisted convolution 5 3. Hermite and Laguerre functions 6 3.1. Hermite polynomials 6 3.2. Laguerre polynomials 9 3.3. Special Hermite functions 9 4. Group Fourier transform of radial functions on the Heisenberg group 12 References 13 1. Fourier transform on Rn We start by presenting some standard properties of the Euclidean Fourier transform; see for example [6] and [4]. Given f ∈ L1(Rn), we define its Fourier transform by setting Z fb(ξ) = e−ix·ξf(x)dx. Rn n ih·ξ If for h ∈ R we let (τhf)(x) = f(x + h), then it follows that τdhf(ξ) = e fb(ξ). Now for suitable f the inversion formula Z f(x) = (2π)−n eix·ξfb(ξ)dξ, Rn holds and we see that the Fourier transform decomposes a function into a continuous sum of characters (eigenfunctions for translations). If A is an orthogonal matrix and ξ is a column vector then f[◦ A(ξ) = fb(Aξ) and from this it follows that the Fourier transform of a radial function is again radial. In particular the Fourier transform −|x|2/2 n of Gaussians take a particularly nice form; if G(x) = e , then Gb(ξ) = (2π) 2 G(ξ). In general the Fourier transform of a radial function can always be explicitly expressed in terms of a Bessel 1 2 NEIL LYALL transform; if g(x) = g0(|x|) for some function g0, then Z ∞ n 2−n n−1 gb(ξ) = (2π) 2 g0(r)(r|ξ|) 2 J n−2 (r|ξ|)r dr, 0 2 where J n−2 is a Bessel function.
    [Show full text]
  • Lecture Notes: Harmonic Analysis
    Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 40506-0027 August 14, 2009 ii Contents Preface vii 1 The Fourier transform on L1 1 1.1 Definition and symmetry properties . 1 1.2 The Fourier inversion theorem . 9 2 Tempered distributions 11 2.1 Test functions . 11 2.2 Tempered distributions . 15 2.3 Operations on tempered distributions . 17 2.4 The Fourier transform . 20 2.5 More distributions . 22 3 The Fourier transform on L2. 25 3.1 Plancherel's theorem . 25 3.2 Multiplier operators . 27 3.3 Sobolev spaces . 28 4 Interpolation of operators 31 4.1 The Riesz-Thorin theorem . 31 4.2 Interpolation for analytic families of operators . 36 4.3 Real methods . 37 5 The Hardy-Littlewood maximal function 41 5.1 The Lp-inequalities . 41 5.2 Differentiation theorems . 45 iii iv CONTENTS 6 Singular integrals 49 6.1 Calder´on-Zygmund kernels . 49 6.2 Some multiplier operators . 55 7 Littlewood-Paley theory 61 7.1 A square function that characterizes Lp ................... 61 7.2 Variations . 63 8 Fractional integration 65 8.1 The Hardy-Littlewood-Sobolev theorem . 66 8.2 A Sobolev inequality . 72 9 Singular multipliers 77 9.1 Estimates for an operator with a singular symbol . 77 9.2 A trace theorem. 87 10 The Dirichlet problem for elliptic equations. 91 10.1 Domains in Rn ................................ 91 10.2 The weak Dirichlet problem . 99 11 Inverse Problems: Boundary identifiability 103 11.1 The Dirichlet to Neumann map . 103 11.2 Identifiability . 107 12 Inverse problem: Global uniqueness 117 12.1 A Schr¨odingerequation .
    [Show full text]
  • CONVERGENCE of FOURIER SERIES in Lp SPACE Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. F
    CONVERGENCE OF FOURIER SERIES IN Lp SPACE JING MIAO Abstract. The convergence of Fourier series of trigonometric functions is easy to see, but the same question for general functions is not simple to answer. We study the convergence of Fourier series in Lp spaces. This result gives us a criterion that determines whether certain partial differential equations have solutions or not.We will follow closely the ideas from Schlag and Muscalu's Classical and Multilinear Harmonic Analysis. Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. Fej´erkernel and Approximate identity 6 4. Lp convergence of partial sums 9 Appendix A. Proofs of Theorems and Lemma 16 Acknowledgments 18 References 18 1. Fourier Series, Partial Sums, and Dirichlet Kernel Let T = R=Z be the one-dimensional torus (in other words, the circle). We consider various function spaces on the torus T, namely the space of continuous functions C(T), the space of H¨oldercontinuous functions Cα(T) where 0 < α ≤ 1, and the Lebesgue spaces Lp(T) where 1 ≤ p ≤ 1. Let f be an L1 function. Then the associated Fourier series of f is 1 X (1.1) f(x) ∼ f^(n)e2πinx n=−∞ and Z 1 (1.2) f^(n) = f(x)e−2πinx dx: 0 The symbol ∼ in (1.1) is formal, and simply means that the series on the right-hand side is associated with f. One interesting and important question is when f equals the right-hand side in (1.1). Note that if we start from a trigonometric polynomial 1 X 2πinx (1.3) f(x) = ane ; n=−∞ Date: DEADLINES: Draft AUGUST 18 and Final version AUGUST 29, 2013.
    [Show full text]
  • Inner Product on Matrices), 41 a (Closure of A), 308 a ≼ B (Matrix Ordering
    Index A : B (inner product on matrices), 41 K∞ (asymptotic cone), 19 A • B (inner product on matrices), 41 K ∗ (dual cone), 19 A (closure of A), 308 L(X,Y ) (linear maps X → Y ), 312 A . B (matrix ordering), 184 L∞(), 311 A∗(adjoint operator), 313 L p(), 311 BX (open unit ball), 17 λ (Lebesgue measure), 317 C(; X) (continuous maps → X), λmin(B) (minimum eigenvalue of B), 315 142, 184 ∞ Rd →∞ C0 ( ) (space of test functions), 316 liminfn xn (limit inferior), 309 co(A) (convex hull), 314 limsupn→∞ xn (limit superior), 309 cone(A) (cone generated by A), 337 M(A) (space of measures), 318 d D(R ) (space of distribution), 316 NK (x) (normal cone), 19 Dα (derivative with multi-index), 322 O(g(s)) (asymptotic “big Oh”), 204 dH (A, B) (Hausdorff metric), 21 ∂ f (subdifferential of f ), 340 δ (Dirac-δ function), 3, 80, 317 P(X) (power set; set of subsets of X), 20 + δH (A, B) (one-sided Hausdorff (U) (strong inverse), 22 − semimetric), 21 (U) (weak inverse), 22 diam F (diameter of set F), 319 K (projection map), 19, 329 n divσ (divergence of σ), 234 R+ (nonnegative orthant), 19 dµ/dν (Radon–Nikodym derivative), S(Rd ) (space of tempered test 320 functions), 317 domφ (domain of convex function), 54, σ (stress tensor), 233 327 σK (support function), 328 epi f (epigraph), 19, 327 sup(A) (supremum of A ⊆ R), 309 ε (strain tensor), 232 TK (x) (tangent cone), 19 −1 f (E) (inverse set), 308 (u, v)H (inner product), 17 f g (inf-convolution), 348 u, v (duality pairing), 17 graph (graph of ), 21 W m,p() (Sobolev space), 322 Hess f (Hessian matrix), 81
    [Show full text]
  • No Tits Alternative for Cellular Automata
    No Tits alternative for cellular automata Ville Salo∗ University of Turku January 30, 2019 Abstract We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alter- native. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group Z) and the one-sided case (spatially acting monoid N, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional sub- shifts of finite type with positive entropy and dense minimal points, auto- morphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field. 1 Introduction In [52] Jacques Tits proved that if F is a field (with no restrictions on charac- teristic), then a finitely-generated subgroup of GL(n, F ) either contains a free group on two generators or contains a solvable subgroup of finite index. We say that a group G satisfies the Tits alternative if whenever H is a finitely-generated subgroup of G, either H is virtually solvable or H contains a free group with two generators. Whether an infinite group satisfies the Tits alternative is one of the natural questions to ask.
    [Show full text]
  • Survey on Geometric Group Theory
    M¨unster J. of Math. 1 (2008), 73–108 M¨unster Journal of Mathematics urn:nbn:de:hbz:6-43529465833 c M¨unster J. of Math. 2008 Survey on geometric group theory Wolfgang L¨uck (Communicated by Linus Kramer) Abstract. This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. Introduction This survey article on geometric group theory is written by a non-expert who likes geometric group theory and uses it in his own research. It is meant as a service for people who want to receive an impression and read an introduction about the topic and possibly will later pass to more elaborate and specialized survey articles or to actual research articles. There will be no proofs. Except for Theorem 7.4 all results have already appeared in the literature. There is to the author’s knowledge no obvious definition what geometric group theory really is. At any rate the basic idea is to pass from a finitely generated group to the geometry underlying its Cayley graph with the word metric. It turns out that only the large scale geometry is really an invariant of the group itself but that this large scale or coarse geometry carries a lot of information. This leads also to a surprising and intriguing variety of new results and structural insights about groups. A possible explanation for this may be that humans have a better intuition when they think in geometric terms.
    [Show full text]
  • The Plancherel Formula for Parabolic Subgroups
    ISRAEL JOURNAL OF MATHEMATICS, Vol. 28. Nos, 1-2, 1977 THE PLANCHEREL FORMULA FOR PARABOLIC SUBGROUPS BY FREDERICK W. KEENE, RONALD L. LIPSMAN* AND JOSEPH A. WOLF* ABSTRACT We prove an explicit Plancherel Formula for the parabolic subgroups of the simple Lie groups of real rank one. The key point of the formula is that the operator which compensates lack of unimodularity is given, not as a family of implicitly defined operators on the representation spaces, but rather as an explicit pseudo-differential operator on the group itself. That operator is a fractional power of the Laplacian of the center of the unipotent radical, and the proof of our formula is based on the study of its analytic properties and its interaction with the group operations. I. Introduction The Plancherel Theorem for non-unimodular groups has been developed and studied rather intensively during the past five years (see [7], [11], [8], [4]). Furthermore there has been significant progress in the computation of the ingredients of the theorem (see [5], [2])--at least in the case of solvable groups. In this paper we shall give a completely explicit description of these ingredients for an interesting family of non-solvable groups. The groups we consider are the parabolic subgroups MAN of the real rank 1 simple Lie groups. As an intermediate step we also obtain the Plancherel formula for the exponential solvable groups AN (see also [6]). In that case our results are more extensive than those of [5] since in addition to the "infinitesimal" unbounded operators we also obtain explicitly the "global" unbounded operator on L2(AN).
    [Show full text]
  • Functional Analysis an Elementary Introduction
    Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Functional Analysis An Elementary Introduction https://doi.org/10.1090//gsm/156 Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 46-01, 46Cxx, 46N20, 35Jxx, 35Pxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-156 Library of Congress Cataloging-in-Publication Data Haase, Markus, 1970– Functional analysis : an elementary introduction / Markus Haase. pages cm. — (Graduate studies in mathematics ; volume 156) Includes bibliographical references and indexes. ISBN 978-0-8218-9171-1 (alk. paper) 1. Functional analysis—Textbooks. 2. Differential equations, Partial—Textbooks. I. Title. QA320.H23 2014 515.7—dc23 2014015166 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society.
    [Show full text]
  • Lightning Talk - UNCG Mathematics Summer School 2019
    Lightning Talk - UNCG Mathematics Summer School 2019 Jenny Beck UNC Greensboro June 25, 2019 Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro Theorem (Tits) A finitely generated linear group is not virtually solvable if and only if it contains a free subgroup. I Historical Context: The Banach-Tarski Paradox and The Von Neumann Conjecture Von Neumann Conjecture (false) A group is non-amenable if and only if it contains a free subgroup I In short, sometimes it pays to be wrong, especially if you’re John Von Neumann.
    [Show full text]
  • Groups of Piecewise Projective Homeomorphisms
    Groups of piecewise projective homeomorphisms Nicolas Monod1 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Edited by Gregory A. Margulis, Yale University, New Haven, CT, and approved February 5, 2013 (received for review October 22, 2012) The group of piecewise projective homeomorphisms of the line The main result of this article is the following, for which we provides straightforward torsion-free counterexamples to the so- introduce a method for proving amenability. called von Neumann conjecture. The examples are so simple that many additional properties can be established. Theorem 1. The group H(A) is nonamenable if A ≠ Z. The next result is a sequacious generalization of the corre- free groups | paradoxical decomposition | von Neumann problem sponding theorem of Brin–Squier (16) about piecewise affine transformations, and we claim no originality. n 1924, Banach and Tarski (1) accomplished a rather para- Idoxical feat. They proved that a solid ball can be decomposed Theorem 2. The group H does not contain any nonabelian free sub- fi group. Thus, H(A) inherits this property for any subring A < R. into ve pieces, which are then moved around and reassembled = R in such a way as to obtain two balls identical to the original one Thus, already H H( ) itself is a counterexample to the von Neumann conjecture. Writing H(A) as the directed union of its (1). This wellnigh miraculous duplication was based on Haus- fi dorff’s (2) 1914 work. nitely generated subgroups, we deduce Corollary 3. In his 1929 study of the Hausdorff–Banach–Tarski paradox, Corollary 3.
    [Show full text]
  • Methods of Applied Mathematics
    Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999{2001, 2004{2005, 2007{2008 (corrected version) by T. Arbogast and J. Bona. Contents Chapter 1. Preliminaries 5 1.1. Elementary Topology 5 1.2. Lebesgue Measure and Integration 13 1.3. Exercises 23 Chapter 2. Normed Linear Spaces and Banach Spaces 27 2.1. Basic Concepts and Definitions. 27 2.2. Some Important Examples 34 2.3. Hahn-Banach Theorems 43 2.4. Applications of Hahn-Banach 48 2.5. The Embedding of X into its Double Dual X∗∗ 52 2.6. The Open Mapping Theorem 53 2.7. Uniform Boundedness Principle 57 2.8. Compactness and Weak Convergence in a NLS 58 2.9. The Dual of an Operator 63 2.10. Exercises 66 Chapter 3. Hilbert Spaces 73 3.1. Basic Properties of Inner-Products 73 3.2. Best Approximation and Orthogonal Projections 75 3.3. The Dual Space 78 3.4. Orthonormal Subsets 79 3.5. Weak Convergence in a Hilbert Space 86 3.6. Exercises 87 Chapter 4. Spectral Theory and Compact Operators 89 4.1. Definitions of the Resolvent and Spectrum 90 4.2. Basic Spectral Theory in Banach Spaces 91 4.3. Compact Operators on a Banach Space 93 4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99 4.5. Compact Self-Adjoint Operators on a Hilbert Space 104 4.6. The Ascoli-Arzel`aTheorem 107 4.7. Sturm Liouville Theory 109 4.8.
    [Show full text]
  • Arxiv:1702.07955V1 [Math.GR]
    ABOUT VON NEUMANN’S PROBLEM FOR LOCALLY COMPACT GROUPS FRIEDRICH MARTIN SCHNEIDER Abstract. We note a generalization of Whyte’s geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups. 1. Introduction In his seminal article [18] von Neumann introduced the concept of amenabil- ity for groups in order to explain why the Banach-Tarski paradox occurs only for dimension greater than two. He proved that a group containing an isomorphic copy of the free group F2 on two generators is not amenable. The converse, i.e., the question whether every non-amenable group would have a subgroup being iso- morphic to F2, was first posed in print by Day [5], but became known as the von Neumann problem (or sometimes von Neumann-Day problem). The original ques- tion has been answered in the negative by Ol’ˇsanski˘ı[19]. However, there are very interesting positive solutions to variants of the von Neumann problem in different settings: a geometric solution by Whyte [27], a measure-theoretic solution by Ga- boriau and Lyons [9] and its generalization to locally compact groups by Gheysens and Monod [14], as well as a Baire category solution by Marks and Unger [13]. Whyte’s geometric version reads as follows. Theorem 1.1 (Theorem 6.2 in [27]).
    [Show full text]