One-parameter families of extreme discrete groups for Jørgensen’s inequality

2000 ◽  
pp. 271-287 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Discrete Groups ◽  
Jørgensen’S Inequality

scholarly journals Chebyshev polynomials and inequalities for Kleinian groups

2021 ◽  
Author(s):  
Hala Alaqad ◽  
Jianhua Gong ◽  
Gaven Martin
Keyword(s):  
Chebyshev Polynomials ◽  
Kleinian Groups ◽  
Necessary Condition ◽  
Complex Numbers ◽  
Discrete Groups ◽  
Low Dimensional Topology ◽  
Principal Character ◽  
Group A ◽  
Low Dimensional ◽  
Jørgensen’S Inequality

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

Distribution of Chromosomes into Discrete Groups in Colchicine-Induced C-Metaphases of Barley

Caryologia ◽  
1985 ◽  
Vol 38 (2) ◽  
pp. 195-206
Author(s):  
J.-P. Zhao ◽  
D. Davidson
Keyword(s):  
Discrete Groups

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals Clebsch–Gordan coefficients of discrete groups in subgroup bases

2018 ◽  
Vol 33 (10) ◽  
pp. 1850055
Author(s):  
Gaoli Chen
Keyword(s):  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Conjugate ◽  
Discrete Groups

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

Drawing Our Fish in the Sand: Secret Biblical Allusions in the Music of U2

2011 ◽  
Vol 19 (2) ◽  
pp. 181-222 ◽  
Author(s):  
Deane Galbraith
Keyword(s):  
Popular Music ◽  
Discrete Groups ◽  
Rock Band ◽  
Semantic Indeterminacy ◽  
Biblical Allusions

AbstractConfronted with a popular music subculture which is predominantly antipathetic to Christianity, the charismatic-evangelical members of rock band U2 double code their lyrics in such a manner that Christian references are hidden from mainstream listeners and media while being readily recognizable to their Christian fans. The device of allusion is especially amenable to this end, as the meaning of an allusion can only be considered by a reader or listener who possesses the requisite competency in respect of the evoked text(s). Through their utilization of biblical allusions, U2 therefore construct two different, perhaps even irreconcilable, groups of listeners—a knowledgeable Christian in-group and an unknowledgeable non-Christian out-group. With detailed reference to U2's songs, this paper examines the covert tendencies of allusion and the manner by which it is able to engage the listener's intertextual imagination. The paper also distinguishes a secret or hidden allusion from a generic allusion on pragmatic and socio-cultural grounds, and demonstrates the potential of secret allusions to increase semantic indeterminacy. Lastly, the paper examines some examples of the reception of the U2 song 'Magnificent' which demonstrate the effectiveness of U2's secret biblical allusions in creating two largely discrete groups of listeners.

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