An Introduction to Combinatorial and Geometric Group Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On Abelian Subgroups of Finitely Generated Metabelian
J. Group Theory 16 (2013), 695–705 DOI 10.1515/jgt-2013-0011 © de Gruyter 2013 On abelian subgroups of finitely generated metabelian groups Vahagn H. Mikaelian and Alexander Y. Olshanskii Communicated by John S. Wilson To Professor Gilbert Baumslag to his 80th birthday Abstract. In this note we introduce the class of H-groups (or Hall groups) related to the class of B-groups defined by P. Hall in the 1950s. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely generated metabelian groups. Hall groups allow us to give a negative answer to G. Baumslag’s conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups. 1 Introduction The subject of our note goes back to the paper of P. Hall [7], which established the properties of abelian normal subgroups in finitely generated metabelian and abelian-by-polycyclic groups. Let B be the class of all abelian groups B, where B is an abelian normal subgroup of some finitely generated group G with polycyclic quotient G=B. It is proved in [7, Lemmas 8 and 5.2] that B H, where the class H of countable abelian groups can be defined as follows (in the present paper, we will call the groups from H Hall groups). By definition, H H if 2 (1) H is a (finite or) countable abelian group, (2) H T K; where T is a bounded torsion group (i.e., the orders of all ele- D ˚ ments in T are bounded), K is torsion-free, (3) K has a free abelian subgroup F such that K=F is a torsion group with trivial p-subgroups for all primes except for the members of a finite set .K/. -
Compression of Vertex Transitive Graphs
Compression of Vertex Transitive Graphs Bruce Litow, ¤ Narsingh Deo, y and Aurel Cami y Abstract We consider the lossless compression of vertex transitive graphs. An undirected graph G = (V; E) is called vertex transitive if for every pair of vertices x; y 2 V , there is an automorphism σ of G, such that σ(x) = y. A result due to Sabidussi, guarantees that for every vertex transitive graph G there exists a graph mG (m is a positive integer) which is a Cayley graph. We propose as the compressed form of G a ¯nite presentation (X; R) , where (X; R) presents the group ¡ corresponding to such a Cayley graph mG. On a conjecture, we demonstrate that for a large subfamily of vertex transitive graphs, the original graph G can be completely reconstructed from its compressed representation. 1 Introduction The complex networks that describe systems in nature and society are typ- ically very large, often with hundreds of thousands of vertices. Examples of such networks include the World Wide Web, the Internet, semantic net- works, social networks, energy transportation networks, the global economy etc., (see e.g., [16]). Given a graph that represents such a large network, an important problem is its lossless compression, i.e., obtaining a smaller size representation of the graph, such that the original graph can be fully restored from its compressed form. We note that, in general, graphs are incompressible, i.e., the vast majority of graphs with n vertices require (n2) bits in any representation. This can be seen by a simple counting argument. There are 2n¢(n¡1)=2 labelled, undirected graphs, and at least 2n¢(n¡1)=2=n! unlabelled, undirected graphs. -
Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields. -
Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a -
Affine Springer Fibers and Affine Deligne-Lusztig Varieties
Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system. -
Pretty Theorems on Vertex Transitive Graphs
Pretty Theorems on Vertex Transitive Graphs Growth For a graph G a vertex x and a nonnegative integer n we let B(x; n) denote the ball of radius n around x (i.e. the set u V (G): dist(u; v) n . If G is a vertex transitive graph then f 2 ≤ g B(x; n) = B(y; n) for any two vertices x; y and we denote this number by f(n). j j j j Example: If G = Cayley(Z2; (0; 1); ( 1; 0) ) then f(n) = (n + 1)2 + n2. f ± ± g 0 f(3) = B(0, 3) = (1 + 3 + 5 + 7) + (5 + 3 + 1) = 42 + 32 | | Our first result shows a property of the function f which is a relative of log concavity. Theorem 1 (Gromov) If G is vertex transitive then f(n)f(5n) f(4n)2 ≤ Proof: Choose a maximal set Y of vertices in B(u; 3n) which are pairwise distance 2n + 1 ≥ and set y = Y . The balls of radius n around these points are disjoint and are contained in j j B(u; 4n) which gives us yf(n) f(4n). On the other hand, the balls of radius 2n around ≤ the points in Y cover B(u; 3n), so the balls of radius 4n around these points cover B(u; 5n), giving us yf(4n) f(5n). Combining our two inequalities yields the desired bound. ≥ Isoperimetric Properties Here is a classical problem: Given a small loop of string in the plane, arrange it to maximize the enclosed area. -
Dynamics for Discrete Subgroups of Sl 2(C)
DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]). -
The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup!1)
transactions of the american mathematical society Volume 150, July, 1970 THE SUBGROUPS OF A FREE PRODUCT OF TWO GROUPS WITH AN AMALGAMATED SUBGROUP!1) BY A. KARRASS AND D. SOLITAR Abstract. We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the inter- section of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann- Neumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G. 1. Introduction. In the case of a free product G = A * B, the structure of a subgroup H may be described as follows (see [9, 4.3]): there exist double coset representative systems {Da}, {De} for G mod (H, A) and G mod (H, B) respectively, and there exists a set of elements t\, t2,.. -
An Ascending Hnn Extension
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 11, November 2006, Pages 3131–3136 S 0002-9939(06)08398-5 Article electronically published on May 18, 2006 AN ASCENDING HNN EXTENSION OF A FREE GROUP INSIDE SL2 C DANNY CALEGARI AND NATHAN M. DUNFIELD (Communicated by Ronald A. Fintushel) Abstract. We give an example of a subgroup of SL2 C which is a strictly ascending HNN extension of a non-abelian finitely generated free group F .In particular, we exhibit a free group F in SL2 C of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of π1(M)inSL2 C with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group F we construct is actually free. 1. Introduction Suppose φ: F → F is an injective homomorphism from a group F to itself. The associated HNN extension H = G, t tgt−1 = φ(g)forg ∈ G is said to be ascending; this extension is called strictly ascending if φ is not onto. In [DS], Drutu and Sapir give examples of residually finite 1-relator groups which are not linear; that is, they do not embed in GLnK for any field K and dimension n. -
Stabilizers of Lattices in Lie Groups
Journal of Lie Theory Volume 4 (1994) 1{16 C 1994 Heldermann Verlag Stabilizers of Lattices in Lie Groups Richard D. Mosak and Martin Moskowitz Communicated by K. H. Hofmann Abstract. Let G be a connected Lie group with Lie algebra g, containing a lattice Γ. We shall write Aut(G) for the group of all smooth automorphisms of G. If A is a closed subgroup of Aut(G) we denote by StabA(Γ) the stabilizer of Γ n n in A; for example, if G is R , Γ is Z , and A is SL(n;R), then StabA(Γ)=SL(n;Z). The latter is, of course, a lattice in SL(n;R); in this paper we shall investigate, more generally, when StabA(Γ) is a lattice (or a uniform lattice) in A. Introduction Let G be a connected Lie group with Lie algebra g, containing a lattice Γ; so Γ is a discrete subgroup and G=Γ has finite G-invariant measure. We shall write Aut(G) for the group of all smooth automorphisms of G, and M(G) = α Aut(G): mod(α) = 1 for the group of measure-preserving automorphisms off G2 (here mod(α) is thegcommon ratio measure(α(F ))/measure(F ), for any measurable set F G of positive, finite measure). If A is a closed subgroup ⊂ of Aut(G) we denote by StabA(Γ) the stabilizer of Γ in A, in other words α A: α(Γ) = Γ . The main question of this paper is: f 2 g When is StabA(Γ) a lattice (or a uniform lattice) in A? We point out that the thrust of this question is whether the stabilizer StabA(Γ) is cocompact or of cofinite volume, since in any event, in all cases of interest, the stabilizer is discrete (see Prop. -
The Free Product of Groups with Amalgamated Subgroup Malnorrnal in a Single Factor
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 127 (1998) 119-136 The free product of groups with amalgamated subgroup malnorrnal in a single factor Steven A. Bleiler*, Amelia C. Jones Portland State University, Portland, OR 97207, USA University of California, Davis, Davis, CA 95616, USA Communicated by C.A. Weibel; received 10 May 1994; received in revised form 22 August 1995 Abstract We discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. In 1971, Karrass and Solitar showed that when the amalgamating subgroup is malnormal in both factors, the global group cannot be two-generator. When the amalgamating subgroup is malnormal in a single factor, the global group may indeed be two-generator. If so, we show that either the non-malnormal factor contains a torsion element or, if not, then there is a generating pair of one of four specific types. For each type, we establish a set of relations which must hold in the factor B and give restrictions on the rank and generators of each factor. @ 1998 Published by Elsevier Science B.V. All rights reserved. 0. Introduction Baumslag introduced the term malnormal in [l] to describe a subgroup that intersects each of its conjugates trivially. Here we discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. -
ADELIC VERSION of MARGULIS ARITHMETICITY THEOREM Hee Oh 1. Introduction Let R Denote the Set of All Prime Numbers Including
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis’s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one ∞ and set Q∞ = R. Let S be any subset of R. For each p ∈ S, let Gp be a connected semisimple adjoint Qp-group without any Qp-anisotropic factors and Dp ⊂ Gp(Qp) be a compact open subgroup for almost all finite prime p ∈ S. Let (GS , Dp) denote the restricted topological product of Gp(Qp)’s, p ∈ S with respect to Dp’s. Note that if S is finite, (GS , Dp) = Qp∈S Gp(Qp). We show that if Pp∈S rank Qp (Gp) ≥ 2, any irreducible lattice in (GS , Dp) is a rational lattice. We also present a criterion on the collections Gp and Dp for (GS , Dp) to admit an irreducible lattice. In addition, we describe discrete subgroups of (GA, Dp) generated by lattices in a pair of opposite horospherical subgroups. 1. Introduction Let R denote the set of all prime numbers including the infinite prime ∞ and Rf the set of finite prime numbers, i.e., Rf = R−{∞}. We set Q∞ = R. For each p ∈ R, let Gp be a non-trivial connected semisimple algebraic Qp-group and for each p ∈ Rf , let Dp be a compact open subgroup of Gp(Qp). The adele group of Gp, p ∈ R with respect to Dp, p ∈ Rf is defined to be the restricted topological product of the groups Gp(Qp) with respect to the distinguished subgroups Dp.