DOCSLIB.ORG

2.4 Direct Products and Finitely Generated Abelian Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2.4 Direct Products and Finitely Generated Abelian Groups 2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS31 2.4 Direct Products and Finitely Generated Abelian Groups 2.4.1 Direct Products 4 Def 2.43. The Cartesian product of sets S1,S2, ··· ,Sn is the set of all ordered n-tuples (a1, ··· , an) where ai ∈ Si for i = 1, 2 ··· , n. n Y Si = S1 × S2 × · · · × Sn i=1 := {(a1, a2, ··· , an) | ai ∈ Si for i = 1, 2, ··· , n}. Caution: The notation (a1, a2, ··· , an) here has different meaning from the cycle notation in permutation groups. Thm 2.44. If G1,G2, ··· ,Gn are groups with the corresponding multiplica- Qn tions, then i=1 Gi is a group under the following multiplication: (a1, a2, ··· , an)(b1, b2, ··· , bn) := (a1b1, a2b2, ··· , anbn) Qn for (a1, a2, ··· , an) and (b1, b2, ··· , bn) in i=1 Gi. Proof. Qn 1. Closed: ai ∈ Gi, bi ∈ Gi ⇒ aibi ∈ Gi ⇒ (a1b1, ··· , anbn) ∈ i=1 Gi. 2. Associativity: By the associativity in each Gi. 3. Identity: Let ei be the identity in Gi. Then e := (e1, e2, ··· , en) is the identity of G. −1 −1 4. Inverse: The inverse of (a1, ··· , an) is (a1 , ··· , an ). Ex 2.45. R2, R3 are abelian groups. (Every finite dimensional real vector space is an abelian group that is iso- morphic to certain Rn.) Ex 2.46. Z2 = {(a, b) | a, b ∈ Z} is a subgroup of R2. Qn Qn (If Hi ≤ Gi for i = 1, ··· , n, then i=1 Hi ≤ i=1 Gi). 41st HW: 1, 2, 14, 34, 46 32 CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS Ex 2.47. If ei is the identity of Gi for i = 1, ··· , n. Then n Y Gj '{e1} × {e2} × · · · × {ej−1} × Gj × {ej+1} × · · · × {en} ≤ Gi. i=1 Qn So every Gj is isomorphic to a subgroup of i=1 Gi for j = 1, ··· , n. Ex 2.48. The Klein 4-group V is isomorphic to Z2 ×Z2, which is not cyclic. Ex 2.49. Z2 × Z3 ' Z6 is cyclic, and (1, 1) is a generator. Thm 2.50. Let m and n be positive integers. 1. If gcd(m, n) = 1 (i.e. m and n are relative prime), then Zm × Zn is cyclic and is isomorphic to Zmn, and (1, 1) is a generator of Zm × Zn. 2. If gcd(m, n) 6= 1, then Zm × Zn is not cyclic. Proof. 5 1. Suppose gcd(m, n) = 1. The sum of k copies of (1, 1) is equal to (k mod m, k mod n). The smallest positive integer k that makes (k mod m, k mod n) = (0, 0) is k = mn because m and n are rel- ative prime. So the cyclic subgroup of (1, 1) in Zm × Zn has the order mn = |Zm ×Zn|. So (1, 1) generates Zm ×Zn. It implies that Zm ×Zn is a cyclic group isomorphic to Zmn. 2. Suppose gcd(m, n) = d 6= 1. Given (a, b) ∈ Zm × Zn, the sum of k copies of (a, b) is (ka mod m, kb mod n). Notice that both m and n mn n m divide d = m · d = d · n. Then mn mn ( a mod m, b mod n) = (0, 0). d d So the cyclic subgroup generated by (a, b) has an order no more than mn d < mn = |Zm × Zn|. Thus Zm × Zn can not be generated by any one of its elements, which implies that Zm × Zn is not cyclic. Qn Cor 2.51. The group i=1 Zmi is cyclic and is isomorphic to Zm1m2···mn if and only if the numbers mi for i = 1, ··· , n are pairwise relative prime, that is, the gcd of any two of them is 1. 51st HW: 1, 2, 14, 34, 46 2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS33 Proof. (For reference only) If mi for i = 1, ··· , n are pairwise relative prime, then n n ! n ! Y Y Y Zmi ' Zm1m2 Zmi ' Zm1m2m3 Zmi '···' Zm1m2···mn . i=1 i=3 i=4 Qn Qn Conversely, if i=1 Zmi ' Zm1m2···mn , then i=1 Zmi contains a subgroup isomorphic to Zmi × Zmj for 1 ≤ i < j ≤ n. Every subgroup of the cyclic Qn group i=1 Zmi ' Zm1m2···mn is cyclic. Hence Zmi × Zmj is cyclic. So gcd(mi, mj) = 1 for 1 ≤ i < j ≤ n. Cor 2.52. 6 If a positive integer n is factorized as a product of powers of distinct prime numbers: n1 n2 nr n = (p1) (p2) ··· (pr) , then n n n Zn ' Z(p1) 1 × Z(p2) 2 × · · · × Z(pr) r . n Note: Each Z(pi) i can not be further decomposed into a product of proper nontrivial subgroups. 2 2 Ex 2.53. (Z5) Z52 . The group (Z5) is not cyclic, but Z52 = Z25 is cyclic. Def 2.54. The least common multiple (abbreviated lcm) of positive integers r1, r2, ··· , rn is the smallest positive integer that is a multiple of each ri for i = 1, 2, ··· , n. If M is a multiple of ri for i = 1, ··· , n, then M is a multiple of lcm(r1, ··· , rn). Qn Thm 2.55. Let (a1, a2, ··· , an) ∈ i=1 Gi. If ai is of finite order ri in Gi, Qn then the order of (a1, a2, ··· , an) in i=1 Gi is lcm(r1, r2, ··· , rn). Proof. k k k k (a1, a2, ··· , an) = (a1, a2, ··· , an). So m m m (a1 , a2 , ··· , an ) = (e1, e2, ··· , en) if and only if m is a multiple of ri for i = 1, 2, ··· , n, if and only if m is a multiple of lcm(r1, r2, ··· , rn). The smallest such positive integer is lcm(r1, r2, ··· , rn). So the order of (a1, a2, ··· , an) is lcm(r1, r2, ··· , rn). Ex 2.56. The order of (8, 4, 10) in Z12 × Z60 × Z24. 62nd HW: 13, 15, 32, 47, 50 34 CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS 2.4.2 The Structure of Finitely Generated Abelian Groups Thm 2.57 (Fundamental Theorem of Finitely Generated Abelian Groups). Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups in the form r r r Z(p1) 1 × Z(p2) 2 × · · · × Z(pn) n × Z × Z × · · · × Z where the pi are primes, not necessarily distinct, and the ri are positive integers. The direct product is unique except for possible rearrangement of the factors; that is, the number (Betti number of G) of factor Z is unique r and the prime powers (pi) i are unique. The proof (omitted) is challenging. In particular, every finite abelian group is isomorphic to a product of finite cyclic groups. (Caution: finite group 6= finitely generated group) Ex 2.58. Decompose the following three groups: 2 2 Z4 × Z12 × Z15 ' Z22 × (Z22 × Z3) × (Z3 × Z5) ' (Z22 ) × (Z3) × Z5, 2 2 Z3 × Z4 × Z60 ' Z3 × Z22 × (Z22 × Z3 × Z5) ' (Z22 ) × (Z3) × Z5, 2 Z20 × Z36 ' (Z22 × Z5) × (Z22 × Z32 ) ' (Z22 ) × Z32 × Z5. ∼ Therefore, Z4 × Z12 × Z15 = Z3 × Z4 × Z60 Z20 × Z36. Ex 2.59 (Ex 11.13, p.109). Find all abelian groups, up to isomorphism, of order 360. 2.4.3 Applications Def 2.60. A group G is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise G is indecomposable. Thm 2.61. The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime. Thm 2.62. If G is a finite abelian group, then G has a subgroup of order m for every m that divides |G|. Ex 2.63. If m is a square-free integer, then every abelian group of order m is isomorphic to Zm and is cyclic. 2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS35 2.4.4 Homework, II-11, p.110-p.113 1st 1, 2, 14, 34, 46 2nd 13, 15, 32, 47, 50 3rd 18, 26, 36, 53 (opt) 29, 37, 38, 39, 52, 54.
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/.
    [Show full text]
  • Finitely Generated Groups Are Universal
    Finitely Generated Groups Are Universal Matthew Harrison-Trainor Meng-Che Ho December 1, 2017 Abstract Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure, of any kind, there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences. 1 Introduction Whenever we have a structure with interesting computability-theoretic properties, it is natural to ask whether such examples can be found within particular classes. While one could try to adapt the proof within the new class, it is often simpler to try and code the original structure into a structure in the given class. It has long been known that for certain classes, such as graphs, this is always possible. Hirschfeldt, Khoussainov, Shore, and Slinko [HKSS02] proved that classes of graphs, partial orderings, lattices, integral domains, and 2-step nilpotent groups are \complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations".
    [Show full text]
  • Decomposing Some Finitely Generated Groups Into Free Products With
    Decomposing some finitely generated groups into free products with amalgamation 1 V.V. Benyash-Krivets Introduction . We shall say that a group G is a non-trivial free product with amalgamation if ¡ ¡ G = G1 A G2, where G1 = A = G2 (see [1]). Wall [2] has posed the following question: Which one-relator groups are non-trivial free products with amalgamation? £ Let G = ¢ g , . , g R = = R = 1 be a group with m generators and n 1 m | 1 · · · n relations such that def G = m n ¤ 2. It is proved in [4] that G is a non-trivial − free product with amalgamation. In particular, if G is a group with m ¤ 3 generators and one relation, then G is a non-trivial free product with amalgamation. A case of groups with two generators and one relation is more complicated. For example, the free −1 −1 £ abelian group G = ¢ a, b [a, b] = 1 of rank 2, where [a, b] = aba b , obviously, is | not a non-trivial free product with amalgamation. Other examples are given by groups −1 n £ Gn = ¢ a, b aba = b . For any n, the group Gn is solvable and using results from [3], | ¡ it is easy to show that for n = 1 the group G is not decomposable into a non-trivial − n free product with amalgamation. The following conjecture was stated in [4]. m £ ¤ Conjecture 1 Let G = ¢ a, b R (a, b) = 1 , m 2, be a group with two generators | and one relation with torsion. Then G is a non-trivial free product with amalgamation.
    [Show full text]
  • Lecture 4, Geometric and Asymptotic Group Theory
    Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n.
    [Show full text]
  • Arxiv:1711.08410V2
    LATTICE ENVELOPES URI BADER, ALEX FURMAN, AND ROMAN SAUER Abstract. We introduce a class of countable groups by some abstract group- theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ2- Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possi- ble lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature. 1. Introduction 1.1. Motivation and background. Let G be a locally compact second countable group, hereafter denoted lcsc1. Such a group carries a left-invariant Radon measure unique up to scalar multiple, known as the Haar measure. A subgroup Γ < G is a lattice if it is discrete, and G/Γ carries a finite G-invariant measure; equivalently, if the Γ-action on G admits a Borel fundamental domain of finite Haar measure. If G/Γ is compact, one says that Γ is a uniform lattice, otherwise Γ is a non- uniform lattice. The inclusion Γ ֒→ G is called a lattice embedding. We shall also say that G is a lattice envelope of Γ. The classical examples of lattices come from geometry and arithmetic. Starting from a locally symmetric manifold M of finite volume, we obtain a lattice embedding π1(M) ֒→ Isom(M˜ ) of its fundamental group into the isometry group of its universal covering via the action by deck transformations.
    [Show full text]
  • Finitely Generated Subgroups of Lattices in Psl2 C
    FINITELY GENERATED SUBGROUPS OF LATTICES IN PSL2 C YAIR GLASNER, JUAN SOUTO, AND PETER STORM Abstract. Let Γ be a lattice in PSL2(C). The pro-normal topology on Γ is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup ∆ < Γ is closed in the pro-normal topology. As a corollary we deduce that if H is a maximal subgroup of a lattice in PSL2(C) then either H is finite index or H is not finitely generated. 1. Introduction In most infinite groups, it is virtually impossible to understand the lattice of all subgroups. Group theorists therefore focus their attention on special families of subgroups such as finite index subgroups, normal subgroups, or finitely generated subgroups. This paper will study the family of finitely generated subgroups of lattices in PSL2(C). Recall that lattices in PSL2(C) are the fundamental groups of finite volume hyperbolic 3-orbifolds. For these groups we establish a connection between finitely generated subgroups and normal subgroups. This connection is best expressed in the following topological terms. If a family of subgroups N is invariant under conjugation and satisfies the con- dition, (1) for all N1,N2 ∈ N ∃ N3 ∈ N such that N3 ≤ N1 ∩ N2 then one can define an invariant topology in which the given family of groups constitutes a basis of open neighborhoods for the identity element. The most famous example is the pro-finite topology, which is obtained by taking N to be the family of (normal) finite index subgroups.
    [Show full text]
  • Relative Subgroup Growth and Subgroup Distortion
    Relative Subgroup Growth and Subgroup Distortion Tara C. Davis, Alexander Yu. Olshanskii October 29, 2018 Abstract We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embed- dings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generted subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group. 1 Introduction and Background We will use the notation that for a group G with finite generating set X, and for an element g G, then g X represents the word length of ∈ | | the element g with respect to the generating set X. For this paper, N = 1, 2,..., . For r N we use the notation that BG(r) represents the ball { } ∈ in G centered at the identity with radius r. That is, BG(r) = g G : { ∈ g X r . | | ≤ } Definition 1.1. Let G be a group finitely generated by a set X with any subset H. The growth function of H in G is G g : N N : r # h H : h X r = #(BG(r) H). H → → { ∈ | | ≤ } ∩ arXiv:1212.5208v1 [math.GR] 20 Dec 2012 The growth functions of subsets and other functions of this type be- come asymptotic invariants independent of the choice of a finite set X if one takes them up to a certain equivalence (see the definitions of , , ∼ ∼ and in Section 3.) ≈ The growth of a subgroup H with respect to a finite set of generators of a bigger group G is called the relative growth of H in G and denoted by G G gH .
    [Show full text]
  • 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.
    [Show full text]
  • FINITELY GENERATED COMMUTATIVE SEMIGROUPS by D
    FINITELY GENERATED COMMUTATIVE SEMIGROUPS by D. B. McALISTER and L. O'CARROLL (Received 17 January, 1969; revised 11 September, 1969) Since all the semigroups considered in this paper are commutative, we shall use the terms " semigroup " and " group " where we actually mean " commutative semigroup " and " commutative group ". Some basic results from the theory of semigroups are required and will be used without explicit mention; these results may be found in [1, § 4.3]. We shall denote the additive semigroups of integers, positive integers, negative integers, positive rationals by Z, Z+, Z", Q+ respectively. 1. Introduction. In his book The theory of finitely generated commutative semigroups, L. Redei has carried out a deep analysis of the congruences on finitely generated free semi- groups. As a consequence of this theory, one can, in principle, construct all finitely generated semigroups. However, Re"dei's theory does not give rise to a structure theorem for such semi- groups. In this paper we consider the problem of giving a description of finitely generated semigroups in the spirit of the so-called " Fundamental theorem of finitely generated groups " [3, Theorems 10.3, 10.4]. We shall state this result as Theorem 1.1 because it serves as a model for our results; taken with our results, it also shows where the analogy between groups and semigroups breaks down. THEOREM 1.1. Let G be a finitely generated group. (I) Gx Z" x Ffor some non-negative integer n and some finite group F. The integer n is unique and the group F is unique up to isomorphism.
    [Show full text]
  • Problems on the Geometry of Finitely Generated Solvable Groups
    Problems on the geometry of finitely generated solvable groups Benson Farb∗ and Lee Moshery February 28, 2000 Contents 1 Introduction 2 2 Dioubina’s examples 5 3 Nilpotent groups and Pansu’s Theorem 5 4 Abelian-by-cyclic groups: nonpolycyclic case 8 5 Abelian-by-cyclic groups: polycyclic case 11 6 Next steps 14 ∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. ySupported in part by NSF grant DMS 9504946. 1 1 Introduction Our story begins with a theorem of Gromov, proved in 1980. Theorem 1 (Gromov’s Polynomial Growth Theorem [Gr1]). Let G be any finitely generated group. If G has polynomial growth then G is virtu- ally nilpotent, i.e. G has a finite index nilpotent subgroup. Gromov’s theorem inspired the more general problem (see, e.g. [GH1, BW, Gr1, Gr2]) of understanding to what extent the asymptotic geometry of a finitely generated solvable group determines its algebraic structure. One way in which to pose this question precisely is via the notion of quasi- isometry. A (coarse) quasi-isometry between metric spaces is a map f : X Y ! such that, for some constants K; C; C0 > 0: 1. 1 d (x ; x ) C d (f(x ); f(x )) Kd (x ; x )+C for all x ; x K X 1 2 − ≤ Y 1 2 ≤ X 1 2 1 2 2 X. 2. The C0-neighborhood of f(X) is all of Y . X and Y are quasi-isometric if there exists a quasi-isometry X Y . Note ! that the quasi-isometry type of a metric space X does not depend on any bounded subset of X; hence the term “asymptotic”.
    [Show full text]
  • Lectures on Lattices
    LECTURES ON LATTICES TSACHIK GELANDER 1. Lecture 1, a brief overview on the theory of lattices Let G be a locally compact group equipped with a left Haar measure µ, i.e. a Borel regular measure which is finite on compact, positive on open and invariant under left multiplications | by Haar's theorem such µ exists and is unique up to normalization. The group G is called unimodular if µ is also right invariant, or equivalently if it is symmetric in the sense that µ(A) = µ(A−1) for every measurable set A. Note that G is compact iff µ(G) < 1. For example: • Compact groups, Nilpotent groups and Perfect groups are unimodular. • The group of affine transformations of the real line is not unimodular. A closed subgroup H ≤ G is said to be co-finite if the quotient space G=H admits a non-trivial finite G invariant measure. A lattice in G is a co-finite discrete subgroup. A discrete subgroup Γ ≤ G is a lattice iff it admits a finite measure fundamental domain, i.e. a measurable set Ω of finite measure which form a set of right cosets representatives for Γ in G. We shall denote Γ ≤L G to express that Γ is a lattice in G. Exercise 1. If G admits a lattice then it is unimodular. We shall say that a closed subgroup H ≤ G is uniform if it is cocompact, i.e. if G=H is compact. Exercise 2. A uniform discrete subgroup Γ ≤ G is a lattice. Note that if G = SL2(R) and H is the Boral subgroup of upper triangular matrices, then H is uniform but not co-finite.
    [Show full text]
  • Discrete Subgroups of Lie Groups
    Discrete subgroups of Lie groups Michael Kapovich (University of California, Davis), Gregory Margulis (Yale University), Gregory Soifer (Bar Ilan University), Dave Witte Morris (University of Lethbridge) December 9, 2019–December 13, 2019 1 Overview of the Field Recent years have seen a great deal of progress in our understanding of “thin” subgroups, which are dis- crete matrix groups that have infinite covolume in their Zariski closure. (The subgroups of finite covolume are called “lattices” and, generally speaking, are much better understood.) Traditionally, thin subgroups are required to be contained in arithmetic lattices, which is natural in the context of number-theoretic and al- gorithmic problems but, from the geometric or dynamical viewpoint, is not necessary. Thin subgroups have deep connections with number theory (see e.g. [6, 7, 8, 18, 22]), geometry (e.g. [1, 14, 16, 38]), and dynamics (e.g. [3, 23, 27, 29]). The well-known “Tits Alternative” [44] (based on the classical “ping-pong argument” of Felix Klein) constructs free subgroups of any matrix group that is not virtually solvable. (In most cases, it is easy to arrange that the resulting free group is thin.) Sharpening and refining this classical construction is a very active and fruitful area of research that has settled numerous old problems. For instance, Breuillard and Gelander [11] proved a quantitative form of the Tits Alternative, which shows that the generators of a free subgroup can be chosen to have small word length, with respect to any generating set of the ambient group. Kapovich, Leeb and Porti [26] provided a coarse-geometric proof of the existence of free subgroups that are Anosov.
    [Show full text]