Embedding and Nonembedding Results for R. Thompson's Group V and Related Groups
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The Wreath Product Principle for Ordered Semigroups Jean-Eric Pin, Pascal Weil
The wreath product principle for ordered semigroups Jean-Eric Pin, Pascal Weil To cite this version: Jean-Eric Pin, Pascal Weil. The wreath product principle for ordered semigroups. Communications in Algebra, Taylor & Francis, 2002, 30, pp.5677-5713. hal-00112618 HAL Id: hal-00112618 https://hal.archives-ouvertes.fr/hal-00112618 Submitted on 9 Nov 2006 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. November 12, 2001 The wreath product principle for ordered semigroups Jean-Eric Pin∗ and Pascal Weily [email protected], [email protected] Abstract Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions of the operations L ! La and L ! LaA∗. Next we turn to concatenation hierarchies. It was shown by Straubing that the n-th level Bn of the dot-depth hierarchy is the variety Vn ∗LI, where LI is the variety of locally trivial semigroups and Vn is the n-th level of the Straubing-Th´erien hierarchy. -
Proper Actions of Wreath Products and Generalizations
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 6, June 2012, Pages 3159–3184 S 0002-9947(2012)05475-4 Article electronically published on February 9, 2012 PROPER ACTIONS OF WREATH PRODUCTS AND GENERALIZATIONS YVES CORNULIER, YVES STALDER, AND ALAIN VALETTE Abstract. We study stability properties of the Haagerup Property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product. 1. Introduction A countable group is Haagerup if it admits a metrically proper isometric action on a Hilbert space. Groups with the Haagerup Property are also known, after Gromov, as a-T-menable groups as they generalize amenable groups. However, they include a wide variety of non-amenable groups: for example, free groups are a-T-menable; more generally, so are groups having a proper isometric action either on a CAT(0) cubical complex, e.g. any Coxeter group, or on a real or complex hyperbolic symmetric space, or on a product of several such spaces; this includes the Baumslag-Solitar group BS(p, q), which acts properly by isometries on the product of a tree and a real hyperbolic plane. A nice feature about Haagerup groups is that they satisfy the strongest form of the Baum-Connes conjecture, namely the conjecture with coefficients [HK]. The Haagerup Property appears as an obstruction to Kazhdan’s Property T and its weakenings such as the relative Property T. Namely, a countable group G has Kazhdan’s Property T if every isometric action on a Hilbert space has bounded orbits; more generally, if G is a countable group and X a subset, the pair (G, X) has the relative Property T if for every isometric action of G on a Hilbert space, the “X-orbit” of 0, {g · 0|g ∈ X} is bounded. -
Deformations of Functions on Surfaces by Isotopic to the Identity Diffeomorphisms
DEFORMATIONS OF FUNCTIONS ON SURFACES BY ISOTOPIC TO THE IDENTITY DIFFEOMORPHISMS SERGIY MAKSYMENKO Abstract. Let M be a smooth compact connected surface, P be either the real line R or the circle S1, f : M ! P be a smooth map, Γ(f) be the Kronrod-Reeb graph of f, and O(f) be the orbit of f with respect to the right action of the group D(M) of diffeomorphisms of M. Assume that at each of its critical point the map f is equivalent to a homogeneous polynomial R2 ! R without multiple factors. In a series of papers the author proved that πnO(f) = πnM for n ≥ 3, π2O(f) = 0, and that π1O(f) contains a free abelian subgroup of finite index, however the information about π1O(f) remains incomplete. The present paper is devoted to study of the group G(f) of automorphisms of Γ(f) induced by the isotopic to the identity diffeomorphisms of M. It is shown that if M is orientable and distinct from sphere and torus, then G(f) can be obtained from some finite cyclic groups by operations of finite direct products and wreath products from the top. As an application we obtain that π1O(f) is solvable. 1. Introduction Study of groups of automorphisms of discrete structures has a long history. One of the first general results was obtained by A. Cayley (1854) and claims that every finite group G of order n is a subgroup of the permutation group of a set consisting of n elements, see also E. Nummela [35] for extension this fact to topological groups and discussions. -
LMS – EPSRC Durham Symposium
LMS – EPSRC Durham Symposium Anthony Byrne Grants and Membership Administrator 12th July 2016, Durham The work of the LMS for mathematics The charitable aims of the Society: Funding the advancement of mathematical knowledge Encouraging mathematical research and collaboration ’, George Legendre Celebrating mathematical 30 Pieces achievements Publishing and disseminating mathematical knowledge Advancing and promoting mathematics The attendees of the Young Researchers in Mathematics Conference 2015, held at Oxford Historical Moments of the London Mathematical Society 1865 Foundation of LMS at University College London George Campbell De Morgan organised the first meeting, and his father, Augustus De Morgan became the 1st President 1865 First minute book list of the 27 original members 1866 LMS moves to Old Burlington House, Piccadilly J.J. Sylvester, 2nd President of the Society. 1866 Julius Plûcker Thomas Hirst Plûcker Collection of boxwood models of quartic surfaces given to Thomas Archer Hirst, Vice- President of LMS, and donated to the Society 1870 Move to Asiatic Society, 22 Albemarle Street William Spottiswoode, President 1874 Donation of £1,000 from John William Strutt (Lord Rayleigh) Generous donation enabled the Society to publish volumes of the Proceedings of the London Mathematical Society. J.W. Strutt (Lord Rayleigh), LMS President 1876-78 1881 First women members Charlotte Angas Scott and Christine Ladd 1884 First De Morgan medal awarded to Arthur Cayley 1885 Sophie Bryant First woman to have a paper published in LMS Proceedings 1916 Return to Burlington House the home of LMS until 1998 1937 ACE ’s Automatic Turing LMS Proceedings, 1937 Computing Engine, published Alan Turing’s first paper 1950 On Computable Numbers 1947 Death of G.H. -
Uniform Simplicity of Groups with Proximal Action
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 4, Pages 110–130 (September 6, 2017) http://dx.doi.org/10.1090/btran/18 UNIFORM SIMPLICITY OF GROUPS WITH PROXIMAL ACTION SWIATOS´ LAW R. GAL AND JAKUB GISMATULLIN, WITH AN APPENDIX BY NIR LAZAROVICH Abstract. We prove that groups acting boundedly and order-primitively on linear orders or acting extremely proximally on a Cantor set (the class in- cluding various Higman-Thomson groups; Neretin groups of almost automor- phisms of regular trees, also called groups of spheromorphisms; the groups of quasi-isometries and almost-isometries of regular trees) are uniformly simple. Explicit bounds are provided. Contents 1. Introduction 110 2. Burago-Ivanov-Polterovich method 115 3. Bounded actions on ordered sets 116 4. Proximality, primitivity, and double-transitivity 118 5. Extremely proximal actions on a Cantor set and uniform simplicity 120 6. Groups almost acting on trees 123 7. Appendix by Nir Lazarovich: Simplicity of AI(Tq)andQI(Tq) 127 Acknowledgments 129 References 129 1. Introduction Let Γ be a group. It is called N-uniformly simple if for every nontrivial f ∈ Γ and nontrivial conjugacy class C ⊂ Γ the element f istheproductofatmost N elements from C±1. A group is uniformly simple if it is N-uniformly simple for some natural number N. Uniformly simple groups are called sometimes, by other authors, groups with finite covering number or boundedly simple groups (see, e.g., [15, 19, 22]). We call Γ boundedly simple if N is allowed to depend on C. The purpose of this paper is to prove results on uniform simplicity, in particular Received by the editors June 14, 2016, and, in revised form, February 27, 2017. -
Fourier Bases for Solving Permutation Puzzles
Fourier Bases for Solving Permutation Puzzles Horace Pan Risi Kondor Department of Computer Science Department of Computer Science, Statistics University of Chicago University of Chicago [email protected] Flatiron Institute [email protected] Abstract Puzzle use pattern databases (Korf and Taylor, 1996; Korf, 1997; Kociemba, n.d.; Felner et al., 2004), which effectively store the distance of various partially solved Traditionally, permutation puzzles such as the puzzles to the solution state. Korf (1997) famously Rubik’s Cube were often solved by heuristic solved instances of the Rubik’s Cube using a pattern search like A∗-search and value based rein- database of all possible corner cube configurations and forcement learning methods. Both heuristic various edge configurations and has since improved search and Q-learning approaches to solving upon the result with an improved hybrid heuristic these puzzles can be reduced to learning a search (Bu and Korf, 2019). heuristic/value function to decide what puz- zle move to make at each step. We propose Machine learning based methods have become popular learning a value function using the irreducible alternatives to classical planning and search approaches representations basis (which we will also call for solving these puzzles. Brunetto and Trunda (2017) the Fourier basis) of the puzzle’s underlying treat the problem of solving the Rubik’s cube as a group. Classical Fourier analysis on real val- supervised learning task by training a feed forward neu- ued functions tells us we can approximate ral network on 10 million cube samples, whose ground smooth functions with low frequency basis truth distances to the goal state were computed by functions. -
Entropy and Isoperimetry for Linear and Non-Linear Group Actions
Entropy and Isoperimetry for Linear and non-Linear Group Actions. Misha Gromov June 11, 2008 Contents 1 Concepts and Notation. 3 incr max 1.1 Combinatorial Isoperimetric Profiles I◦, I◦ and I◦ . 3 1.2 I◦ for Subgroups, for Factor Groups and for Measures on Groups. 4 max 1.3 Boundaries |∂| , ∂µ and the Associated Isoperimetric Profiles. 5 1.4 G·m-Boundary and Coulhon-Saloff-Coste Inequality . 6 1.5 Linear Algebraic Isoperimetric Profiles I∗ of Groups and Algebras. 7 max 1.6 The boundaries |∂| , |∂|µ and |∂|G∗m ............... 9 1.7 I∗ For Decaying Functions. 10 1.8 Non-Amenability of Groups and I∗-Linearity Theorems by Elek and Bartholdi. 11 1.9 The Growth Functions G◦ and the Følner Functions F◦ of Groups and F∗ of Algebras. 12 1.10 Multivariable Følner Functions, P◦-functions and the Loomis- Whitney Inequality. 14 1.11 Filtered Boundary F∂ and P∗-Functions. 15 1.12 Summary of Results. 17 2 Non-Amenability of Groups and Strengthened non-Amenability of Group Algebras. 19 2.1 l2-Non-Amenability of the Complex Group Algebras of non- Amenable groups. 19 2.2 Geometric Proof that Applies to lp(Γ). 20 2.3 F- Non-Amenability of the Group Algebras of Groups with Acyclic Families of Subgroups. 21 3 Isoperimetry in the Group Algebras of Ordered Groups. 24 3.1 Triangular Systems and Bases. 24 3.2 Domination by Ordering and Derivation of the Linear Algebraic Isoperimetric Inequality from the Combinatorial One. 24 3.3 Examples of Orderable Groups and Corollaries to The Equiva- lence ◦ ⇔ ∗.............................. -
The Chameleon Groups of Richards J. Thompson : Automorphisms and Dynamics
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. MATTHEW G. BRIN The chameleon groups of Richards J. Thompson : automorphisms and dynamics Publications mathématiques de l’I.H.É.S., tome 84 (1996), p. 5-33 <http://www.numdam.org/item?id=PMIHES_1996__84__5_0> © Publications mathématiques de l’I.H.É.S., 1996, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE CHAMELEON GROUPS OF RICHARD J. THOMPSON: AUTOMORPHISMS AND DYNAMICS by MATTHEW G. BRIN CONTENTS Part I. Background........................................................................... 6 0. Introduction ............................................................................. Q 1. Statements, history and outline............................................................. 8 1.1. Definitions and statements ............................................................ 8 1.2. Outline of the paper ................................................................. 10 1:3. Prior and related work ............................................................... 10 1.4. Other literature on -
Wreath Products in Permutation Group Theory
Wreath products in permutation group theory Csaba Schneider Universidade de Bras´ılia, 19–21 February 2019 NOTES AND BOOKS These notes: Available online The book: Permutation Groups and Cartesian Decompositions Other recomended literature: 1. Peter Cameron, Permutation groups. 2. Dixon & Mortimer, Permutation groups. OUTLINE OF LECTURES I the wreath product construction; II the imprimitive action of a wreath product; III the primitive action of a wreath product; IV groups that preserve cartesian decompositions; V twisted wreath products. THE WREATH PRODUCT CONSTRUCTION Input: 1. A group G; 2. a permutation group H 6 S`. ` Set B = G . An element h 2 H induces an αh 2 Aut(B): (g1;:::; g`)αh = (g1h−1 ;:::; g`h−1 ) for all gi 2 G; h 2 H: The map ' : H ! Aut(B); h 7! αh is a homomorphism. Example Let ` = 4, h = (1; 2; 3) 2 S4: (1;2;3) (g1; g2; g3; g4)α(1;2;3) = (g1; g2; g3; g4) = (g3; g1; g2; g4): THE WREATH PRODUCT Define ` W = G o H = B o' H = G o' H: A generic element of W can be written as (g1;:::; g`)h with gi 2 G; h 2 H: The multiplication in W: 0 0 0 0 0 h−1 0 [(g1;:::; g`)h][(g1;:::; g`)h ] = (g1;:::; g`)(g1;:::; g`) hh = 0 0 0 0 0 0 (g1;:::; g`)(g1h;:::; g`h)hh = (g1g1h;:::; g`g`h)hh : Terminology: 1. B is the base group of W and B E W. 2. H is the top group of W. THE IMPRIMITIVE ACTION Suppose from now on that G is also a permutation group: G 6 Sym Γ. -
The Banach–Tarski Paradox and Amenability Lecture 25: Thompson's Group F
The Banach{Tarski Paradox and Amenability Lecture 25: Thompson's Group F 30 October 2012 Thompson's groups F , T and V In 1965 Richard Thompson (a logician) defined 3 groups F ≤ T ≤ V Each is finitely presented and infinite, and Thompson used them to construct groups with unsolvable word problem. From the point of view of amenability F is the most important. The group F is not elementary amenable and does not have a subgroup which is free of rank ≥ 2. We do not know if F is amenable. The groups T and V are finitely presented infinite simple groups. The commutator subgroup [F ; F ] of F is finitely presented and ∼ 2 simple and F =[F ; F ] = Z . Definition of F We can define F to be the set of piecewise linear homeomorphisms f : [0; 1] ! [0; 1] which are differentiable except at finitely many a dyadic rationals (i.e. rationals 2n 2 (0; 1)), and such that on intervals of differentiability the derivatives are powers of 2. Examples See sketches of A and B. Lemma F is a subgroup of the group of all homeomorphisms [0; 1] ! [0; 1]. Proof. Let f 2 F have breakpoints 0 = x0 < x1 < ··· < xn = 1. Then f (x) = a1x on [0; x1] where a1 is a power of 2, so f (x1) = a1x1 is a dyadic rational. Hence f (x) = a2x + b2 on [x1; x2] where a2 is a power of 2 and b2 is dyadic rational. By induction f (x) = ai x + bi on [xi−1; xi ] with ai a power of 2 and bi a dyadic rational. -
C 2010 KENDALL NICOLE MAURER ALL RIGHTS RESERVED
c 2010 KENDALL NICOLE MAURER ALL RIGHTS RESERVED MINIMALLY SIMPLE GROUPS AND BURNSIDE’S THEOREM A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Kendall Nicole Maurer May, 2010 MINIMALLY SIMPLE GROUPS AND BURNSIDE’S THEOREM Kendall Nicole Maurer Thesis Approved: Accepted: Advisor Dean of the College Dr. James P. Cossey Dr. Chand K. Midha Faculty Reader Dean of the Graduate School Dr. Jeffrey Riedl Dr. George R. Newkome Faculty Reader Date Dr. Antonio Quesada Department Chair Dr. Joseph Wilder ii ABSTRACT William Burnside’s paqb theorem is a very important result in group theory, which states that any group G of order paqb is solvable. An interesting fact about this the- orem is that it was originally proven using techniques from character theory, another branch of algebra. In fact, it was about seventy years before a group-theoretic proof of Burnside’s theorem was developed through the work of Goldschmidt, Matsuyama, Bender, and other mathematicians. Their approach to proving the theorem was to show that, in essence, minimally simple groups of size paqb do not exist. Our purpose here is to use various techniques from the group-theoretic proof of Burnside’s theorem to establish and prove similar results about minimally simple groups G of arbitrary order. iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my wonderful family and friends, especially my mother and father; my sisters, Michelle and Staci; and Grandma Mikey. You always inspire me and give so much of yourselves. -
On Closed Subgroups of the R. Thompson Group F
On closed subgroups of the R. Thompson group F G. Golan-Polak, M. Sapir∗ Abstract We prove that Thompson’s group F has a subgroup H such that the conjugacy problem in H is undecidable and the membership problem in H is easily decidable. The subgroup H of F is a closed subgroup of F . That is, every function in F which is a piecewise-H function belongs to H. Other interesting examples of closed −→ subgroups of F include Jones’ subgroups F n and Jones’ 3-colorable subgroup F. By a recent result of the first author, all maximal subgroups of F of infinite index are closed. In this paper we prove that if K ≤ F is finitely generated then the closure of K, i.e., the smallest closed subgroup of F which contains K, is finitely generated. We also prove that all finitely generated closed subgroups of F are undistorted in F . In particular, all finitely generated maximal subgroups of F are undistorted in F . 1 Introduction Recall that R. Thompson’s group F is the group of all piecewise-linear homeomorphisms of the interval [0, 1] with finitely many breakpoints, where all breakpoints are finite Z 1 dyadic (i.e., elements of the set [ 2 ] ∩ (0, 1)) and all slopes are integer powers of 2. The group F has a presentation with two generators and two defining relations [13]. Algorithmic problems in F have been extensively studied. It is well known that the word problem in F is decidable in linear time [13, 34, 21] and that the conjugacy problem is decidable in linear time [21, 3, 4].