DOCSLIB.ORG

Discrete Subgroups of Lie Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. In a somewhat different vein, Margulis and Soifer [32] proved that SL(n; Z); n ≥ 4; contains 2 free products of the form Z ?Fk (where Fk is the free group of rank k) for all k ≥ 1. (Answering a question of Platonov and Prasad, this implies that SL(n; Z) has maximal subgroups of infinite index that are not free groups.) Also, it has been shown that SL(n; Z) contains Coxeter groups (and, hence, right-angled Artin groups) when n is sufficiently large. Ping-pong type constructions have also emerged as an important technical tool in other contexts, such as for disproving the invariable generation property (by constructing a thin subgroup that intersects every conjugacy class [19]) and for proving the expander property for Cayley graphs of finite quotients of thin groups [22]. Conversely, there are also obstructions to the existence of thin subgroups. For example, no thin, Zariski- dense subgroup of SL(n; Z) contains SL(3; Z) [45]. Similarly, it has been shown in certain situations that thin, discrete, Zariski-dense subgroups cannot contain a lattice in a maximal unipotent subgroup of the ambi- ent group [5, 35, 46]. The study of certain natural (and properly discontinuous) actions of thin groups is another important line of research. The famous Auslander Conjecture concerns actions of thin groups on affine spaces. (Namely, it is conjectured that if a group acts properly discontinuously and cocompactly on an affine space Rn, then the group is virtually solvable.) Actions on more general geometric spaces (such as flag varieties) are also 1 2 important. While it seems that nothing of interest can be said about such actions in general, a great deal of progress has been made in recent years analyzing actions of thin subgroups that satisfy further restrictions. The Anosov property has been of particular interest (see, for example, [23], [27], and [49]), as well as other forms of strengthening of discreteness, e.g. regularity, which has its origin in [3], see also [27]. This conference brought together a range of specialists whose expertise in order to educate each other in the broad spectrum of techniques and problems in the theory of discrete subgroups of Lie groups. A number of young researchers, including graduate students, actively participated in the conference as well. 2 Open Problems On the first day of the workshop we conducted a problem session in the theory of discrete subgroups of Lie groups. Below are problems collected during and after the workshop. 2.1 Proper actions on affine spaces Let Γ be a finitely generated group, ρ :Γ ! GL(n; R) be the given representation, Z1(Γ; ρ) be the space of ρ-cocycles with values in Rn, i.e. maps n u :Γ ! R ; u(αβ) = u(α) + ρ(α)u(β); α; β 2 Γ: Note that Z1(Γ; ρ) is a finite-dimensional real vector space. Each cocycle u 2 Z1(Γ; ρ) determines an affine n action ρu of Γ on V = R : ρu(γ): x 7! ρ(γ)x + u(γ): 1 n Let C = Cρ ⊂ Z (Γ; ρ) denote the subset consisting of cocycles such that the action ρu of Γ on R is properly discontinuous. Question 2.1 (N. Tholozan). Is C open in Z1(Γ; ρ)? Is it convex? Note that the answer is positive for n = 3, this follows from the results of [21]. Question 2.2 (N. Tholozan). Does Cρ depend continuously on ρ? Here one has to be careful with the topology used on the set of subsets of Z1(Γ; ρ). For closed subsets one uses Shabauty topology. For instance, if the subsets C are open, their complements are closed and, hence, one can interpret the question as of the continuity of the complement with respect to ρ. For a finitely generated group Γ, let P (Γ; n) denote the subset of Hom(Γ; Aff(Rn)) consisting of rep- resentations defining proper actions of Γ on Rn. Let PA(Γ; n) ⊂ P (Γ; n) denote the subset consisting of actions ρu with P -Anosov linear part ρ (for some parabolic subgroup P < GL(n; R)). Question 2.3 (G. Soifer). To which extent P (Γ; n) is open? Note that in general P (Γ; n) is not open even for n = 3, for instance, one can take a rank 2 free group Γ and a representation ρu 2 P (Γ; 3) whose linear part ρ contains unipotent elements. A small perturbation of ρu will yield a representation (with linear part in SO(2; 1)) with nondiscrete linear part, hence, a non-proper affine action. One can also perturb a representation so that the linear part is deformed to a Zariski dense subgroup of SL(3; R), again resulting in a non-proper action. It is known that 3 PA(Γ; 3) \ Hom(Γ; SO(2; 1) n R ) is open in 3 Hom(Γ; SO(2; 1) n R ) (this follows from the results of [21] and stability of Anosov representations). Question 2.4 (G. Soifer). To which extent P (Γ; n) is open in general? 3 Conjecture 2.5 (The Auslander conjecture). If Γ < Aff(Rn) is a subgroup which acts properly discontinu- ously and co-compactly on Rn, then Γ is virtually solvable. Abels, Margulis and Soifer proved the Auslander conjecture for the dimensions n ≤ 6 and observed that the following problem is important for the further progress towards the Auslander conjecture: Conjecture 2.6 (Abels, Margulis, Soifer). Let Γ < O(4; 3) n R7 < Aff(R7) be a subgroup acting properly discontinuously and co-compactly on R7. Then the linear part of Γ is not Zariski dense in SO(4; 3). Question 2.7. Does there exist a properly discontinuous cocompact group of affine transformations isomor- phic to the fundamental group of a closed hyperbolic manifold? Partial progress towards a negative answer to this problem was described at the talk by Suhyoung Choi at the workshop: He proved that such an action cannot exist under certain assumptions on its linear part (strengthening the P -Anosov condition). Question 2.8 (G. Mostow). Suppose that M = Rn=Γ is an affine manifold, such that the linear part of Γ is n in SLn(R). Thus, Γ preserves the standard volume form on R and, hence, M has a canonical volume form as well. Does V ol(M) < 1 imply that M is compact? This question is motivated by Mostow’s theorem that lattices in solvable groups are cocompact. Question 2.9 (G. Soifer). Study subgroups Γ < Aff(Cn) acting properly discontinuously on Cn. More specifically: Question 2.10 (G. Soifer). Does there exist a free nonabelian subgroup Γ < Aff(C3) acting properly discon- 3 3 tinuously on C ? For instance, consider the adjoint action of SL2(C) on C (identified with the Lie algebra of SL2(C)). Consider a generic representation ρ : F2 = ha; bi ! SL(2; C). Does there exist a cocycle 1 3 3 m m u 2 Z (F2; C ) and m > 0 such that the action on C of ha ; b i given by ρu is properly discontinuous? Conjecture 2.11 (Markus Conjecture). Suppose that M is a compact n-dimensional affine manifold whose linear holonomy is in SL(n; R). Is M complete? This conjecture is known when the linear holonomy of M has “discompacity 1” (Y. Carriere, [13]), e.g. when M is a flat Lorentzian manifold, and also for convex affine manifolds of dimension ≤ 5, [25]. Conjecture 2.12 (M. Kapovich). Suppose that M is a compact n-dimensional affine manifold whose linear holonomy is contained in a rank one subgroup of SL(n; R). Is M complete? 2.2 Discrete subgroups of SL(3; R) Question 2.13 (M. Kapovich). 1. Does there exist a discrete a subgroup Γ < SL(3; R) isomorphic to Z2 ? Z and containing only regular diagonalizable elements? 2. Does there exist a discrete subgroup Γ < SL(3; Z) isomorphic to Z2 ? Z? Note that there are known examples, [43], of discrete subgroups Γ < SL(3; R) isomorphic to Z2?Z where Z2 is super-singular: It is generated by three singular diagonalizable matrices A; B; C satisfying ABC = 1.
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]
  • 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.
    [Show full text]
  • 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]).
    [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]
  • Arxiv:Math/0511624V1
    Automorphism groups of polycyclic-by-finite groups and arithmetic groups Oliver Baues ∗ Fritz Grunewald † Mathematisches Institut II Mathematisches Institut Universit¨at Karlsruhe Heinrich-Heine-Universit¨at D-76128 Karlsruhe D-40225 D¨usseldorf May 18, 2005 Abstract We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homo- topy theory. 2000 Mathematics Subject Classification: Primary 20F28, 20G30; Secondary 11F06, 14L27, 20F16, 20F34, 22E40, 55P10 arXiv:math/0511624v1 [math.GR] 25 Nov 2005 ∗e-mail: [email protected] †e-mail: [email protected] 1 Contents 1 Introduction 4 1.1 Themainresults ........................... 4 1.2 Outlineoftheproofsandmoreresults . 6 1.3 Cohomologicalrepresentations . 8 1.4 Applications to the groups of homotopy self-equivalences of spaces 9 2 Prerequisites on linear algebraic groups and arithmetic groups 12 2.1 Thegeneraltheory .......................... 12 2.2 Algebraicgroupsofautomorphisms. 15 3 The group of automorphisms of a solvable-by-finite linear algebraic group 17 3.1 The algebraic structure of Auta(H)................
    [Show full text]
  • Congruence Subgroups of Matrix Groups
    Pacific Journal of Mathematics CONGRUENCE SUBGROUPS OF MATRIX GROUPS IRVING REINER AND JONATHAN DEAN SWIFT Vol. 6, No. 3 BadMonth 1956 CONGRUENCE SUBGROUPS OF MATRIX GROUPS IRVING REINER AND J. D. SWIFT 1. Introduction* Let M* denote the modular group consisting of all integral rxr matrices with determinant + 1. Define the subgroup Gnf of Mt to be the group of all matrices o of Mt for which CΞΞΞO (modn). M. Newman [1] recently established the following theorem : Let H be a subgroup of M% satisfying GmnCZH(ZGn. Then H=Gan, where a\m. In this note we indicate two directions in which the theorem may be extended: (i) Letting the elements of the matrices lie in the ring of integers of an algebraic number field, and (ii) Considering matrices of higher order. 2. Ring of algebraic integers* For simplicity, we restrict our at- tention to the group G of 2x2 matrices (1) . A-C b \c d where α, b, c, d lie in the ring £& of algebraic integers in an algebraic number field. Small Roman letters denote elements of £^, German letters denote ideals in £&. Let G(3l) be the subgroup of G defined by the condition that CΞΞO (mod 91). We shall prove the following. THEOREM 1. Let H be a subgroup of G satisfying (2) G(^)CHCGP) , where (3JΪ, (6)) = (1). Then H^GφW) for some Proof. 1. As in Newman's proof, we use induction on the number of prime ideal factors of 3JL The result is clear for 9Jέ=(l). Assume it holds for a product of fewer than k prime ideals, and let 2Ji==£V«- Qfc (&i^l)> where the O4 are prime ideals (not necessarily distinct).
    [Show full text]
  • Commensurators of Finitely Generated Non-Free Kleinian Groups. 1
    Commensurators of finitely generated non-free Kleinian groups. C. LEININGER D. D. LONG A.W. REID We show that for any finitely generated torsion-free non-free Kleinian group of the first kind which is not a lattice and contains no parabolic elements, then its commensurator is discrete. 57M07 1 Introduction Let G be a group and Γ1; Γ2 < G. Γ1 and Γ2 are called commensurable if Γ1 \Γ2 has finite index in both Γ1 and Γ2 . The Commensurator of a subgroup Γ < G is defined to be: −1 CG(Γ) = fg 2 G : gΓg is commensurable with Γg: When G is a semi-simple Lie group, and Γ a lattice, a fundamental dichotomy es- tablished by Margulis [26], determines that CG(Γ) is dense in G if and only if Γ is arithmetic, and moreover, when Γ is non-arithmetic, CG(Γ) is again a lattice. Historically, the prominence of the commensurator was due in large part to its density in the arithmetic setting being closely related to the abundance of Hecke operators attached to arithmetic lattices. These operators are fundamental objects in the theory of automorphic forms associated to arithmetic lattices (see [38] for example). More recently, the commensurator of various classes of groups has come to the fore due to its growing role in geometry, topology and geometric group theory; for example in classifying lattices up to quasi-isometry, classifying graph manifolds up to quasi- isometry, and understanding Riemannian metrics admitting many “hidden symmetries” (for more on these and other topics see [2], [4], [17], [18], [25], [34] and [37]).
    [Show full text]
  • 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.
    [Show full text]
  • Introduction to Arithmetic Groups What Is an Arithmetic Group? Geometric
    What is an arithmetic group? Introduction to Arithmetic Groups Every arithmetic group is a group of matrices with integer entries. Dave Witte Morris More precisely, SLΓ(n, Z) G : GZ where " ∩ = University of Lethbridge, Alberta, Canada ai,j Z, http://people.uleth.ca/ dave.morris SL(n, Z) nΓ n mats (aij) ∈ ∼ = " × # det 1 $ [email protected] # = # semisimple G SL(n, R) is connected Lie# group , Abstract. This lecture is intended to introduce ⊆ # def’d over Q non-experts to this beautiful topic. Examples For further reading, see: G SL(n, R) SL(n, Z) = ⇒ = 2 2 2 D. W. Morris, Introduction to Arithmetic Groups. G SO(1, n) Isom(x1 x2 xn 1) = = Γ − − ···− + Deductive Press, 2015. SO(1, n)Z ⇒ = http://arxiv.org/src/math/0106063/anc/ subgroup of that has finite index Γ Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 1 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 2 / 12 Γ Geometric motivation Other spaces yield groups that are more interesting. Group theory = the study of symmetry Rn is a symmetric space: y Example homogeneous: x Symmetries of a tessellation (periodic tiling) every pt looks like all other pts. x,y, isometry x ! y. 0 ∀ ∃ x0 reflection through a point y0 (x+ x) is an isometry. = − Assume tiles of tess of X are compact (or finite vol). Then symmetry group is a lattice in Isom(X) G: = symmetry group Z2 Z2 G/ is compact (or has finite volume). = " is cocompact Γis noncocompact Thm (Bieberbach, 1910).
    [Show full text]
  • Congruence Subgroups Undergraduate Mathematics Society, Columbia University
    Congruence Subgroups Undergraduate Mathematics Society, Columbia University S. M.-C. 24 June 2015 Contents 1 First Properties1 2 The Modular Group and Elliptic Curves3 3 Modular Forms for Congruence Subgroups4 4 Sums of Four Squares5 Abstract First we expand the connection between the modular group and ellip- tic curves by defining certain subgroups of the modular group, known as \congruence subgroups", and stating their relation to \enhanced elliptic curves". Then we will carefully define modular forms for congruence sub- groups. Finally, as a neat application, we will use modular forms to count the number of ways an integer can be expressed as a sum of four squares. This talk covers approximately x1.2 and x1.5 of Diamond & Shurman. 1 First Properties Recall the modular group SL2 Z of 2 × 2 integer matrices with determinant one, and its action on the upper half-plane H = fτ 2 C : im(τ) > 0g: aτ + b a b γτ = for γ = 2 SL and τ 2 H: (1) cτ + d c d 2 Z Any subgroup of the modular group inherits this action on the upper half- plane. Our goal is to investigate certain subgroups of the modular group, defined as follows. Let a b a b 1 0 Γ(N) = 2 SL : ≡ mod N (2) c d 2 Z c d 0 1 1 (where we simply reduce each entry mod N). As the kernel of the natural morphism SL2 Z ! SL2 Z=NZ, we see Γ(N) is a normal subgroup of SL2 Z, called the principal congruence subgroup of level N.
    [Show full text]
  • Congruence Subgroup Problem for Algebraic Groups: Old and New Astérisque, Tome 209 (1992), P
    Astérisque A. S. RAPINCHUK Congruence subgroup problem for algebraic groups: old and new Astérisque, tome 209 (1992), p. 73-84 <http://www.numdam.org/item?id=AST_1992__209__73_0> © Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression 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/ CONGRUENCE SUBGROUP PROBLEM FOR ALGEBRAIC GROUPS: OLD AND NEW A. S. RAPINCHUK* Let G C GLn be an algebraic group defined over an algebraic number field K. Let 5 be a finite subset of the set VK of all valuations of K, containing the set V*£ of archimedean valuations. Denote by O(S) the ring of 5-integers in K and by GQ(S) the group of 5-units in G. To any nonzero ideal a C O(S) there corresponds the congruence subgroup Go(s)(*) = {96 G0(s) \ 9 = En (mod a)} , which is a normal subgroup of finite index in GQ(S)- The initial statement of the Congruence Subgroup Problem was : (1) Does any normal subgroup of finite index in GQ(S) contain a suitable congruence subgroup Go(s)(a) ? In fact, it was found by F. Klein as far back as 1880 that for the group SL2(Z) the answer to question (1) is "no".
    [Show full text]
  • [Math.GR] 2 Mar 2002 ∆ Missing”
    COXETER GROUPS, 2-COMPLETION, PERIMETER REDUCTION AND SUBGROUP SEPARABILITY Paul E. Schupp Abstract. We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index. §1. Introduction. Several years ago the author [S2] raised the question of whether or not small cancellation methods could be used to investigate questions about finitely gener- ated subgroups of “sufficiently nice” groups— in particular the solvability of the membership problem and the Howson property. Rips [R] replied “No” by con- ′ 1 structing, for every metric small cancellation condition C n , finitely presented groups which satisfy the condition but which have unsolvable membership prob- lem and are neither Howson nor coherent. However, recent work of McCammond and Wise [M-W] and of Arzhantseva and Olshanskii[A-O] shows that the answer is actually “Yes” in many cases. McCammond and Wise introduced the use of “distributive” small cancellation hypotheses where the condition involves how the generators are distributed among all the defining relators and the ingenious idea of “perimeter reduction” where one counts “what is missing”. In this paper we introduce the idea that if one has a “suitable” subgroup graph, ∆1(H), of a subgroup H, one can construct from it a “complete” subgroup graph, arXiv:math/0203020v1 [math.GR] 2 Mar 2002 ∆2(H) which has most of the properties of subgroup graphs in the case of free groups and which directly reveals desired information about H.
    [Show full text]