Algebraic K–Theory Over the Infinite Dihedral Group:An Algebraic Approach
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Arxiv:1704.05304V2 [Math.GR] 24 Aug 2017 Iergop,Rltvl Yeblcgroups
LINEAR GROUPS, CONJUGACY GROWTH, AND CLASSIFYING SPACES FOR FAMILIES OF SUBGROUPS TIMM VON PUTTKAMER AND XIAOLEI WU Abstract. Given a group G and a family of subgroups F , we consider its classifying space EF G with respect to F . When F = VCyc is the family of virtually cyclic subgroups, Juan- Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by L¨uck–Reich–Rognes–Varisco for the family of cyclic subgroups. Finally, we construct finitely generated groups that exhibit wild inner automorphims but which admit a model for EVCyc(G) whose 0-skeleton is finite. Introduction Given a group G, a family F of subgroups of G is a set of subgroups of G which is closed under conjugation and taking subgroups. We denote by EF (G)a G-CW-model for the classifying space for the family F . The space EF (G) is characterized by the property H that the fixed point set EF (G) is contractible for any H ∈F and empty otherwise. Recall that a G-CW-complex X is said to be finite if it has finitely many orbits of cells. Similarly, X is said to be of finite type if it has finitely many orbits of cells of dimension n for any n. We abbreviate EF (G) by EG for F = VCyc the family of virtually cyclic subgroups, EG for F = F in the family of finite subgroups and EG for F the family consisting only of the trivial subgroup. -
New York Journal of Mathematics Periodic Points on Shifts of Finite Type and Commensurability Invariants of Groups
New York Journal of Mathematics New York J. Math. 21 (2015) 811{822. Periodic points on shifts of finite type and commensurability invariants of groups David Carroll and Andrew Penland Abstract. We explore the relationship between subgroups and the pos- sible shifts of finite type (SFTs) which can be defined on a group. In particular, we investigate two group invariants, weak periodicity and strong periodicity, defined via symbolic dynamics on the group. We show that these properties are invariants of commensurability. Thus, many known results about periodic points in SFTs defined over groups are actually results about entire commensurability classes. Additionally, we show that the property of being not strongly periodic (i.e., the prop- erty of having a weakly aperiodic SFT) is preserved under extensions with finitely generated kernels. We conclude by raising questions and conjectures about the relationship of these invariants to the geometric notions of quasi-isometry and growth. Contents 1. Introduction 811 2. Background 814 2.1. Symbolic dynamics on groups 814 2.2. Quasi-isometry and commensurability of groups 815 3. Some special SFTs on groups 816 3.1. Higher block shift 816 3.2. Products of shifts 817 3.3. Locked shift 818 4. Commensurability and periodicity 818 4.1. SFTs and quotient groups 820 References 821 1. Introduction Given a group G and a finite alphabet A, a shift (over G) is a nonempty compact, G-equivariant subset of the full shift AG, where AG is a topological space regarded as a product of discrete spaces. A shift of finite type (SFT) Received April 6, 2015. -
The Tits Alternative
The Tits Alternative Matthew Tointon April 2009 0 Introduction In 1972 Jacques Tits published his paper Free Subgroups in Linear Groups [Tits] in the Journal of Algebra. Its key achievement was to prove a conjecture of H. Bass and J.-P. Serre, now known as the Tits Alternative for linear groups, namely that a finitely-generated linear group over an arbitrary field possesses either a solvable subgroup of finite index or a non-abelian free subgroup. The aim of this essay is to present this result in such a way that it will be clear to a general mathematical audience. The greatest challenge in reading Tits's original paper is perhaps that the range of mathematics required to understand the theorem's proof is far greater than that required to understand its statement. Whilst this essay is not intended as a platform in which to regurgitate theory it is very much intended to overcome this challenge by presenting sufficient background detail to allow the reader, without too much effort, to enjoy a proof that is pleasing in both its variety and its ingenuity. Large parts of the prime-characteristic proof follow basically the same lines as the characteristic-zero proof; however, certain elements of the proof, particularly where it is necessary to introduce field theory or number theory, can be made more concrete or intuitive by restricting to characteristic zero. Therefore, for the sake of clarity this exposition will present the proof over the complex numbers, although where clarity and brevity are not impaired by considering a step in the general case we will do so. -
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. -
The Minkowski Question Mark Function, Shown in Figure 1
THE MINKOWSKI QUESTION MARK, GL(2;Z) AND THE MODULAR GROUP (EXPOSITORY) LINAS VEPSTAS ABSTRACT. Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in turn describes the structure of the Cantor set. The infinite binary tree can be viewed as a certain subset of the modular group PSL(2;Z). The subset is essentially the dyadic groupoid or dyadic monoid. It provides the natural setting for the symmetry and self-similarity of many frac- tals, including those associated with period-doubling maps, with phase-locking maps, and with various dynamical systems in general. The aim of this text is to provide a simple exposition of the symmetry and its articulation. In the process, this paper attempts to clarify the relationships between a cluster of inter- related ideas from number theory: those surrounding the modular group, elliptic curves and the Cantor Set. It has long been widely known that the modular group PSL(2;Z) (the general linear group of 2 by 2 matrices over the integers) is the symmetric group of elliptic curves. Connections between this group, and the rational numbers are commonly presented in books and coursework; the connection to the dyadic subsets is rarely, if ever, mentioned – this paper attempts to correct this shortcoming. Likewise, the Cantor set plays an important role in many areas of mathematics; among others, it is a superset of the real numbers, and more general a universal cover for compact metric spaces. -
Polycyclic, Metabelian Or Soluble of Type (FP) $ {\Infty} $ Groups With
POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN VITALY ROMAN’KOV Omsk State Dostoevsky University and Omsk State Technical University, 644077, Omsk, Russia, e-mail:[email protected] Abstract Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a boolean algebra. Then G is virtually abelian. Every soluble biautomatic group is virtually abelian. 1 Introduction The topic of this paper is two important concepts: rational sets and biauto- matic structure. We study finitely generated soluble groups G such that the class Rat(G) of all rational subsets of G is a boolean algebra. We conjecture that every such group is virtually abelian. Note that every finitely generated virtually abelian group satisfies to this property. We confirm this conjecture in the case where G is a polycyclic, metabelian or soluble group of type FP∞. This conjecture remains open in general case. It appeared that the notion FP∞ helps to prove by the way that every soluble biautomatic group is virtually abelian. Thus, we give answer to known question posed in [1]. We provide full proofs of four theorems attributed to Bazhenova, a former student of the author, stating that some natural assumptions imply that a sol- uble group is virtually abelian. The original proofs was given by her in collab- arXiv:1511.00223v1 [math.GR] 1 Nov 2015 oration with the author more than 14 years ago and had never been published. -
Quasi-Isometry Rigidity of Groups
Quasi-isometry rigidity of groups Cornelia DRUT¸U Universit´e de Lille I, [email protected] Contents 1 Preliminaries on quasi-isometries 2 1.1 Basicdefinitions .................................... 2 1.2 Examplesofquasi-isometries ............................. 4 2 Rigidity of non-uniform rank one lattices 6 2.1 TheoremsofRichardSchwartz . .. .. .. .. .. .. .. .. 6 2.2 Finite volume real hyperbolic manifolds . 8 2.3 ProofofTheorem2.3.................................. 9 2.4 ProofofTheorem2.1.................................. 13 3 Classes of groups complete with respect to quasi-isometries 14 3.1 Listofclassesofgroupsq.i. complete. 14 3.2 Relatively hyperbolic groups: preliminaries . 15 4 Asymptotic cones of a metric space 18 4.1 Definition,preliminaries . .. .. .. .. .. .. .. .. .. 18 4.2 Asampleofwhatonecandousingasymptoticcones . 21 4.3 Examplesofasymptoticconesofgroups . 22 5 Relatively hyperbolic groups: image from infinitely far away and rigidity 23 5.1 Tree-graded spaces and cut-points . 23 5.2 The characterization of relatively hyperbolic groups in terms of asymptotic cones 25 5.3 Rigidity of relatively hyperbolic groups . 27 5.4 More rigidity of relatively hyperbolic groups: outer automorphisms . 29 6 Groups asymptotically with(out) cut-points 30 6.1 Groups with elements of infinite order in the center, not virtually cyclic . 31 6.2 Groups satisfying an identity, not virtually cyclic . 31 6.3 Existence of cut-points in asymptotic cones and relative hyperbolicity . 33 7 Open questions 34 8 Dictionary 35 1 These notes represent a slightly modified version of the lectures given at the summer school “G´eom´etriesa ` courbure n´egative ou nulle, groupes discrets et rigidit´es” held from the 14-th of June till the 2-nd of July 2004 in Grenoble. -
Lectures on Quasi-Isometric Rigidity Michael Kapovich 1 Lectures on Quasi-Isometric Rigidity 3 Introduction: What Is Geometric Group Theory? 3 Lecture 1
Contents Lectures on quasi-isometric rigidity Michael Kapovich 1 Lectures on quasi-isometric rigidity 3 Introduction: What is Geometric Group Theory? 3 Lecture 1. Groups and spaces 5 1. Cayley graphs and other metric spaces 5 2. Quasi-isometries 6 3. Virtual isomorphisms and QI rigidity problem 9 4. Examples and non-examples of QI rigidity 10 Lecture 2. Ultralimits and Morse lemma 13 1. Ultralimits of sequences in topological spaces. 13 2. Ultralimits of sequences of metric spaces 14 3. Ultralimits and CAT(0) metric spaces 14 4. Asymptotic cones 15 5. Quasi-isometries and asymptotic cones 15 6. Morse lemma 16 Lecture 3. Boundary extension and quasi-conformal maps 19 1. Boundary extension of QI maps of hyperbolic spaces 19 2. Quasi-actions 20 3. Conical limit points of quasi-actions 21 4. Quasiconformality of the boundary extension 21 Lecture 4. Quasiconformal groups and Tukia's rigidity theorem 27 1. Quasiconformal groups 27 2. Invariant measurable conformal structure for qc groups 28 3. Proof of Tukia's theorem 29 4. QI rigidity for surface groups 31 Lecture 5. Appendix 33 1. Appendix 1: Hyperbolic space 33 2. Appendix 2: Least volume ellipsoids 35 3. Appendix 3: Different measures of quasiconformality 35 Bibliography 37 i Lectures on quasi-isometric rigidity Michael Kapovich IAS/Park City Mathematics Series Volume XX, XXXX Lectures on quasi-isometric rigidity Michael Kapovich Introduction: What is Geometric Group Theory? Historically (in the 19th century), groups appeared as automorphism groups of certain structures: • Polynomials (field extensions) | Galois groups. • Vector spaces, possibly equipped with a bilinear form | Matrix groups. -
Coarse Geometry and Finitely Generated Groups
N.J.Towner Coarse geometry and finitely generated groups Bachelor thesis, July 2013 Supervisor: Dr. L.D.J. Taelman Mathematisch Instituut, Universiteit Leiden 2 Contents 0 Conventions 3 1 Introduction 4 2 Length spaces and geodesic spaces 7 3 Coarse geometry and group actions 22 4 Polycyclic groups 30 5 Growth functions of finitely generated groups 38 A Acknowledgements 49 References 50 0 CONVENTIONS 3 0 Conventions 1. We write N for the set of all integers greater than or equal to 0. 2. For sets A and B, A ⊂ B means that every element of A is also an element of B. 3. We use the words \map" and \function" synonymously, with their set-theoretic meaning. 4. For subsets A; B ⊂ R, a map f : A ! B is called increasing if for all x; y 2 A, x > y ) f(x) > f(y). 5. An edge of a graph is a set containing two distinct vertices of the graph. 6. For elements x, y of a group, the commutator [x; y] is defined as xyx−1y−1. 7. For metric spaces (X; dX ), (Y; dY ), we call a map f : X ! Y an 0 0 0 isometric embedding if for all x; x 2 X, dY (f(x); f(x )) = dX (x; x ). We call f an isometry if it is a surjective isometric embedding. 1 INTRODUCTION 4 1 Introduction The topology of a metric space cannot tell us much about its large-scale, long-range structure. Indeed, if we take a metric space (X; d) and for each x; y 2 X we let d0(x; y) = minfd(x; y); 1g then d0 is a metric on X equivalent to d. -
Virtual Properties of 3-Manifolds Dedicated to the Memory of Bill Thurston
Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston Ian Agol ∗ Abstract. We will discuss the proof of Waldhausen's conjecture that compact aspheri- cal 3-manifolds are virtually Haken, as well as Thurston's conjecture that hyperbolic 3- manifolds are virtually fibered. The proofs depend on major developments in 3-manifold topology of the past decades, including Perelman's resolution of the geometrization con- jecture, results of Kahn and Markovic on the existence of immersed surfaces in hyperbolic 3-manifolds, and Gabai's sutured manifold theory. In fact, we prove a more general the- orem in geometric group theory concerning hyperbolic groups acting on CAT(0) cube complexes, concepts introduced by Gromov. We resolve a conjecture of Dani Wise about these groups, making use of the theory that Wise developed with collaborators including Bergeron, Haglund, Hsu, and Sageev as well as the theory of relatively hyperbolic Dehn filling developed by Groves-Manning and Osin. Mathematics Subject Classification (2010). Primary 57M Keywords. hyperbolic, 3-manifold 1. Introduction In Thurston's 1982 Bulletin of the AMS paper Three Dimensional Manifolds, Kleinian groups, and hyperbolic geometry [118], he asked 24 questions which have guided the last 30 years of research in the field. Four of the questions have to do with \virtual" properties of 3-manifolds: • Question 15 (paraphrased): Are Kleinian groups LERF? [76, Problem 3.76 (Hass)] • Question 16: \Does every aspherical 3-manifold have a finite-sheeted cover which is Haken?" This -
On the Quasi-Isometric Classification of Locally Compact Groups
ON THE QUASI-ISOMETRIC CLASSIFICATION OF LOCALLY COMPACT GROUPS YVES CORNULIER Abstract. This (quasi-)survey addresses the quasi-isometry classification of lo- cally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homoge- neous negatively curved manifolds. A main conjecture provides a general descrip- tion; an extended discussion reduces this conjecture to more specific statements. In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compact groups. 19.1. Introduction 19.1.1. Locally compact groups as geometric objects. It has long been well understood in harmonic analysis (notably in the study of unitary representations) that locally compact groups are the natural objects unifying the setting of connected Lie groups and discrete groups. In the context of geometric group theory, this is still far from universal. For a long period, notably in the 70s, this unifying point of view was used essentially by Herbert Abels, and, more occasionally, some other people including Behr, Guivarc'h, Houghton. The considerable influence of Gromov's work paradoxically favored the bipolar point of view discrete vs continuous, although Gromov's ideas were applicable to the setting of locally compact groups and were sometimes stated (especially in [Gr87]) in an even greater generality. If a locally compact group is generated by a compact subset S, it can be endowed with the word length with respect to S, and with the corresponding left-invariant distance. -
Metabelian Groups: Full-Rank Presentations, Randomness And
Metabelian groups: full-rank presentations, randomness and Diophantine problems Albert Garreta, Leire Legarreta, Alexei Miasnikov, Denis Ovchinnikov Abstract We study metabelian groups G given by full rank finite presentations xA | RyM in the variety M of metabelian groups. We prove that G is a product of a free metabelian subgroup of rank maxt0, |A| ´ |R|u and a virtually abelian normal subgroup, and that if |R| ď |A|´ 2 then the Diophantine problem of G is undecidable, while it is decidable if |R| ě |A|. We further prove that if |R| ď |A|´ 1 then in any direct decomposition of G all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely. Contents 1 Introduction 2 1.1 Metabelian groups with full rank presentations . ... 2 1.2 Diophantineproblems ........................ 3 1.3 Random finitely presented groups: the few-relations model.... 4 2 Structural properties of metabelian groups given by full-rank presentations 5 2.1 A variation of the Generalized Freiheissatz . ... 6 2.2 Directdecompositions . .. .. .. .. .. .. 8 3 Diophantine problem 8 3.1 Diophantine problem and e-interpretability . ... 8 3.2 The Diophantine problem in metabelian groups given by full rank arXiv:2006.06371v1 [math.GR] 11 Jun 2020 presentations ............................. 10 4 Random metabelian groups 10 5 Acknowledgements 11 6 Bibliography 12 1 1 Introduction In this paper we study finitely generated metabelian groups G given by full rank finite presentations G “ xa1,...,an | r1,...,rmyM in the variety M of metabelian groups, random metabelian groups in the few relators model, and the Diophantine problem in such groups.