On Property (RD) Certain Discrete Groups
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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]). -
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]). -
Some Groups of Finite Homological Type
Some groups of finite homological type Ian J. Leary∗ M¨ugeSaadeto˘glu† June 10, 2005 Abstract For each n ≥ 0 we construct a torsion-free group that satisfies K. S. Brown’s FHT condition and is FPn, but is not FPn+1. 1 Introduction While working on comparing different notions of Euler characteristic, K. S. Brown introduced a new homological finiteness condition for discrete groups [5, IX.6]. The group G is said to be of finite homological type or FHT if G has finite virtual cohomological dimension, and for every G-module M whose underlying abelian group is finitely generated, the homology groups Hi(G; M) are all finitely generated. If G is FHT , then one may define a ‘na¨ıve Euler characteristic’ for every finite-index subgroup H of G, as the alternating sum of the dimensions of the homology groups of H with rational coefficients. One question that arises is the connection between FHT and the usual homological finiteness conditions FP and FPn, which were introduced by J.-P. Serre [8]. (We shall define these conditions below.) It is easy to see that any group G of type FP is FHT , and one might conjecture that every torsion-free group that is FHT is also of type FP . The aim of this paper is to show that this is not the case. For each n ≥ 0, we exhibit a torsion-free group Gn that is FHT and of type FPn, but that is not of type FPn+1. Our construction is based on R. Bieri’s construction of a group that is FPn but not FPn+1 [2, Prop 2.14]. -
A Theorem on Discrete Groups and Some Consequences of Kazdan's
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 6, 203-207 (1970) A Theorem on Discrete Groups and Some Consequences of Kazdan’s Thesis ROBERT R. KALLMAN* Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by Irving Segal Received June 21, 1969 Let G be a noncompact simple Lie group, and let r be a discrete subgroup of G such that G/P has finite volume. The main result of this paper is that the left ring of P is a Type II1 von Neumann algebra. This result in turn is applied to solve, in some generality, a problem in dynamical systems posed by I. M. Gelfand. INTRODUCTION If G is a locally compact unimodular group, let Y(G) be the von Neumann algebra generated by left translations L(a) (a in G) on L2(G). Let I’ be a discrete subgroup of G, and let p be a measure on G/r which is invariant under the natural action of G. The measure class of p is uniquely determined. If f is in L2(G/F), let (V4f)W = flu-’ - 4. The following problem is of interest in dynamical systems (cf. Gelfand [3]). Let &?(G, r) be the von Neumann algebra on L2(G) @ L2(G/.F) generated by L(a) @ V(u) (u in G) and I @ M(4) (here 4 is in L”(G/r, p) and (&Z(+)f)(x) = ~$(x)f(x), for all f in L2(G/r)). -
Discrete Groups and Simple C*-Algebras 1 Introduction
Discrete groups and simple C*-algebras by Erik Bedos* Department of Mathematics University of Oslo P.O.Box 1053, Blindern 0316 Oslo 3, Norway 1 Introduction Let G denote a discrete group and let us say that G is C* -simple if the reduced group C* -algebra associated with G is simple. We notice im mediately that there is no interest in considering here the full group C* algebra associated with G, because it is simple if and only if G is trivial. Since Powers in 1975 ([26]) proved that all non-abelian free groups are C* simple, the class of C* -simple groups has been considerably enlarged (see [1,2,6,7,12,13,14,16,24] as a sample!), and two important subclasses are the so-called weak Powers groups ([6,13]; see section 4 for definition and ex amples) and the groups of Akemann-Lee type ([1,2]), which are groups possessing a normal non-abelian free subgroup with trivial centralizer. The problem of giving an intrisic characterization of C* -simple groups is still open. It is known that a C* -simple group has no normal amenable subgroup other than the trivial one ([24; proposition 1.6]) and is ICC (since the center of the associated reduced group C* -algebra must be the scalars). One may of course wonder if the converse is true. On the other hand, most C* -simple groups are known to have a unique trace, i.e. the canonical trace on the reduced group C* -algebra· is unique, which naturally raises the problem whether this is always true or not ([13; §2, question (2)]). -
Discreteness Properties of Translation Numbers in Solvable Groups
DISCRETENESS PROPERTIES OF TRANSLATION NUMBERS IN SOLVABLE GROUPS GREGORY R. CONNER Abstract. We define a group to be translation proper if it carries a left- invariant metric in which the translation numbers of the non-torsion elements are nonzero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension, m, is a finite m extension of Z . 1. Introduction The notion of a translation number comes from Riemannian geometry. Suppose M is a compact Riemannian manifold of nonpositive sectional curvature and Mf is its universal cover, enjoying the metric induced from M. Then the covering transformation group acts as isometries on Mf. Under these conditions any covering transformation of Mf fixes a geodesic line. Since a geodesic line is isometric to R, a covering transformation must translate each point of such a fixed geodesic line by a constant amount which is called the translation number of the covering transformation. This notion can be abstracted. Let G be a group which is equipped with a metric, d, which is invariant under left multiplication by elements of G, i.e., d(xy, xz)=d(y,z) ∀ x, y, z ∈ G. We then say that d is a left-invariant group metric k k (or a metric for brevity) on G. Let kk: G −→ Z be defined by x = d(x, 1G). To fix ideas, one may consider a finitely generated group equipped with a word metric. -
3.4 Discrete Groups
3.4 Discrete groups Lemma 3.4.1. Let Λ be a subgroup of R2, and let ~a;~b be two elements of Λ that are linearly independent as vectors in R2. Suppose that the paralleogram spanned by ~a and ~b contains no other elements of Λ other than ~0;~a;~b and ~a +~b. Then Λ is the group Λ = fm~a + n~b j m; n 2 Zg: (3.4) Proof. The plane is tiled up with copies of the parallellogram P spanned by the four vectors ~0;~a;~b;~a+~b. The vertices of these parallellograms are m~a+n~b for some integers m and n. Since Λ is a group all these vertices are in Λ. If Λ is not the group fm~a + n~b j m; n 2 Zg, then there exists an element x 2 Λ not in located at any of the vertices of the parallellograms. Since these parallellograms tile the whole plane there exist m and n such that x is contained in the parallellogram m~a + n~b; (m + 1)~a + n~b; m~a + (n + 1)~b and (m + 1)~a + (n + 1)~b. The element x0 = x − m~a − n~b is in the group Λ, and in the first paralleogram P . By assumption x0 is one of the four vertices of P , and it follows that also x is an vertex of the parallellograms. Hence Λ is the group fm~a + n~b j m; n 2 Zg. Definition 3.4.2. A subgroup Λ of R2 is called a lattice if there exists an > 0 such that every non-zero element ~a 2 Λ has length j~aj > . -
Deforming Discrete Groups Into Lie Groups
Deforming discrete groups into Lie groups March 18, 2019 2 General motivation The framework of the course is the following: we start with a semisimple Lie group G. We will come back to the precise definition. For the moment, one can think of the following examples: • The group SL(n; R) n × n matrices of determinent 1, n • The group Isom(H ) of isometries of the n-dimensional hyperbolic space. Such a group can be seen as the transformation group of certain homoge- neous spaces, meaning that G acts transitively on some manifold X. A pair (G; X) is what Klein defines as “a geometry” in its famous Erlangen program [Kle72]. For instance: n n • SL(n; R) acts transitively on the space P(R ) of lines inR , n n • The group Isom(H ) obviously acts on H by isometries, but also on n n−1 @1H ' S by conformal transformations. On the other side, we consider a group Γ, preferably of finite type (i.e. admitting a finite generating set). Γ may for instance be the fundamental group of a compact manifold (possibly with boundary). We will be interested in representations (i.e. homomorphisms) from Γ to G, and in particular those representations for which the intrinsic geometry of Γ and that of G interact well. Such representations may not exist. Indeed, there are groups of finite type for which every linear representation is trivial ! This groups won’t be of much interest for us here. We will often start with a group of which we know at least one “geometric” representation. -
CHARACTERS of ALGEBRAIC GROUPS OVER NUMBER FIELDS Bachir Bekka, Camille Francini
CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS Bachir Bekka, Camille Francini To cite this version: Bachir Bekka, Camille Francini. CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS. 2020. hal-02483439 HAL Id: hal-02483439 https://hal.archives-ouvertes.fr/hal-02483439 Preprint submitted on 18 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS BACHIR BEKKA AND CAMILLE FRANCINI Abstract. Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G: We deter- mine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is gener- ated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k); this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups. 1. Introduction Let k be a field and G an algebraic group defined over k. When k is a local field (that is, a non discrete locally compact field), the group G = G(k) of k-rational points in G is a locally compact group for the topology induced by k. -
Lattices in Lie Groups
Lattices in Lie groups Dave Witte Morris Lattices in Lie groups 1: Introduction University of Lethbridge, Alberta, Canada http://people.uleth.ca/ dave.morris What is a lattice subgroup? ∼ [email protected] Arithmetic construction of lattices. Abstract Compactness criteria. During this mini-course, the students will learn the Classical simple Lie groups. basic theory of lattices in semisimple Lie groups. Examples will be provided by simple arithmetic constructions. Aspects of the geometric and algebraic structure of lattices will be discussed, and the Superrigidity and Arithmeticity Theorems of Margulis will be described. Dave Witte Morris (U of Lethbridge) Lattices 1: Introduction CIRM, Jan 2019 1 / 18 Dave Witte Morris (U of Lethbridge) Lattices 1: Introduction CIRM, Jan 2019 2 / 18 A simple example. Z2 is a cocompact lattice in R2. 2 2 R is a connected Lie group Replace R with interesting group G. group: (x1,x2) (y1,y2) (x1 y1,x2 y2) + = + + Riemannian manifold (metric space, connected) is a cocompact lattice in G: (discrete subgroup) distance is right-invariant: d(x a, y a) d(x, y) + + = Every pt in G is within bounded distance of . group ops (x y and x) continuous (differentiable) Γ + − g cγ where c is bounded — in cpct set 2 = Z is a discrete subgroup (no accumulation points) compact C G, G C . Γ Every point in R2 is within ∃ ⊆ = G/ is cpct. (gn g ! γn , gnγn g) 2 bdd distance C √2 of Z → Γ ∃{ } → = 2 Z is a cocompact lattice is a latticeΓ in G: Γ Γ ⇒ 2 in R . only require C (or G/ ) to have finite volume. -
R. Ji, C. Ogle, and B. Ramsey. Relatively Hyperbolic
J. Noncommut. Geom. 4 (2010), 83–124 Journal of Noncommutative Geometry DOI 10.4171/JNCG/50 © European Mathematical Society Relatively hyperbolic groups, rapid decay algebras and a generalization of the Bass conjecture Ronghui Ji, Crichton Ogle, and Bobby Ramsey With an Appendix by Crichton Ogle Abstract. By deploying dense subalgebras of `1.G/ we generalize the Bass conjecture in terms of Connes’ cyclic homology theory. In particular, we propose a stronger version of the `1-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy bound property and nilpotent periodicity property, satisfy the `1-Stronger-Bass Conjecture. Moreover, we determine the conjugacy bound for relatively hyperbolic groups and compute the cyclic cohomology of the `1-algebra of any discrete group. Mathematics Subject Classification (2010). 46L80, 20F65, 16S34. Keywords. B-bounded cohomology, rapid decay algebras, Bass conjecture. Introduction Let R be a discrete ring with unit. A classical construction of Hattori–Stallings ([Ha], [St]) yields a homomorphism of abelian groups HS a Tr K0 .R/ ! HH0.R/; a where K0 .R/ denotes the zeroth algebraic K-group of R and HH0.R/ D R=ŒR; R its zeroth Hochschild homology group. Precisely, given a projective module P over R, one chooses a free R-module Rn containing P as a direct summand, resulting in a map n Trace EndR.P / ,! EndR.R / D Mn;n.R/ ! HH0.R/: HS One then verifies the equivalence class of Tr .ŒP / ´ Trace..IdP // in HH0.R/ depends only on the isomorphism class of P , is invariant under stabilization, and sends direct sums to sums. -
Discrete Linear Groups Containing Arithmetic Groups
DISCRETE LINEAR GROUPS CONTAINING ARITHMETIC GROUPS INDIRA CHATTERJI AND T. N. VENKATARAMANA Abstract. If H is a simple real algebraic subgroup of real rank at least two in a simple real algebraic group G, we prove, in a sub- stantial number of cases, that a Zariski dense discrete subgroup of G containing a lattice in H is a lattice in G. For example, we show that any Zariski dense discrete subgroup of SLn(R)(n ≥ 4) which contains SL3(Z) (in the top left hand corner) is commensurable with a conjugate of SLn(Z). In contrast, when the groups G and H are of real rank one, there are lattices ∆ in a real rank one group H embedded in a larger real rank one group G and that extends to a Zariski dense discrete subgroup Γ of G of infinite co-volume. 1. Introduction In this paper, we study special cases of the following problem raised by Madhav Nori: Problem 1 (Nori, 1983). If H is a real algebraic subgroup of a real semi-simple algebraic group G, find sufficient conditions on H and G such that any Zariski dense subgroup Γ of G which intersects H in a lattice in H, is itself a lattice in G. If the smaller group H is a simple Lie group and has real rank strictly greater than one (then the larger group G is also of real rank at least two), we do not know any example when the larger discrete group Γ is not a lattice. The goal of the present paper is to study the following question, related to Nori's problem.