Discrete group
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Maximally stretched laminations on geometrically finite hyperbolic manifolds Open
Let [math] be a discrete group. For a pair [math] of representations of [math] into [math] with [math] geometrically finite, we study the set of [math] –equivariant Lipschitz maps from the real hyperbolic space [math] to itself that have m…
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The Property of Rapid Decay for Discrete Quantum Groups Open
We introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones. We then investigate examples, proving in particular the Property of Rapid Decay for unimodular free qu…
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Discrete complex reflection groups Open
Here are reproduced slightly edited notes of my lectures on the classification of discrete groups generated by complex reflections of Hermitian affine spaces delivered in October of 1980 at the University of Utrecht.
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$$S_3$$ S 3 discrete group as a source of the quark mass and mixing pattern in $$331$$ 331 models Open
\n We propose a model based on the $${SU}(3)_{C}\\otimes SU(3)_{L}\\otimes U(1)_{X}$$ gauge symmetry with an extra $$S_{3}\\otimes Z_{2}\\otimes Z_{4}\\otimes Z_{12}$$ discrete group, which successfully accounts for the SM quark mass and m…
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Localized index and $L^2$-Lefschetz fixed-point formula for orbifolds Open
We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the L²-index of Atiyah, are …
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Δ(27) family symmetry and neutrino mixing Open
The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In …
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Alternative schemes of predicting lepton mixing parameters from discrete flavor and symmetry Open
We suggest two alternative schemes to predict lepton mixing angles as well as $CP$ violating phases from a discrete flavor symmetry group combined with $CP$ symmetry. In the first scenario, the flavor and $CP$ symmetry is broken to the res…
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Quark and lepton mixing patterns from a common discrete flavor symmetry with a generalized symmetry Open
We have studied two approaches to predict quark and lepton mixing patterns from the same flavor symmetry group in combination with CP symmetry. The first approach is based on the residual symmetry $Z_2$ in the charged lepton sector and $Z_…
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LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART I: GENERAL THEORY Open
Let $G$ be a totally disconnected, locally compact group. A closed subgroup of $G$ is locally normal if its normalizer is open in $G$ . We begin an investigation of the structure of the family of closed locally normal subgroups of $G$ . Mo…
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NONCOMMUTATIVE DE LEEUW THEOREMS Open
Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$ . Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subse…
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Lepton sector phases and their roles in flavor and generalized symmetries Open
We study the effects of considering nontrivial unphysical lepton sector phases on the group theoretical properties of the flavor and generalized CP symmetry elements in the case where there are three light, distinct Majorana neutrino speci…
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Continuity of discrete homomorphisms of diffeomorphism groups Open
Let $M$ and $N$ be two closed $C^{\\infty}$ manifolds and let\n$\\text{Diff}_c(M)$ denote the group of $C^{\\infty}$ diffeomorphisms isotopic to\nthe identity. We prove that any (discrete) group homomorphism between\n$\\text{Diff}_c(M)$ an…
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Anosov groups: local mixing, counting and equidistribution Open
Let G be a connected semisimple real algebraic group, and Γ<G a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients ⟨(exptv). f1,f2⟩ in L2(Γ∖G) as t→∞ for a…
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Uniformly recurrent subgroups and the ideal structure of reduced crossed products Open
We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $Γ$, we consider the ab…
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Cheeger constants and L2-Betti numbers Open
We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/\\Gamma$ where $X$ is a contractible Riemannian manifold and $\\Gamma\\lt \\operatorname{Isom}(X)$ is a discrete subgroup, typically with …
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ON GRADED -ALGEBRAS Open
We show that every topological grading of a $C^{\ast }$ -algebra by a discrete abelian group is implemented by an action of the compact dual group.
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Bernoulli disjointness Open
International audience
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A strong hyperbolicity property of locally symmetric varieties Open
We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a we…
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Matrix coefficients, counting and primes for orbits of geometrically finite groups Open
Let G:=\mathrm {SO}(n,1)^\circ and \Gamma(n-1)/2 for n=2,3 and when \delta>n-2 for n\geq 4 , we obtain an effective archimedean counting result for a discrete orbit of \Gamma in a homogeneous space H \backslash G where H is the trivial gro…
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Groups with Spanier–Whitehead duality Open
Building on work by Kasparov, we study the notion of Spanier–Whitehead [math] -duality for a discrete group. It is defined as duality in the [math] -category between two [math] -algebras which are naturally attached to the group, namely th…
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Characterizations of C*-simplicity Open
We characterize discrete groups that are C*-simple, meaning that their reduced C*-algebra is simple. First, we prove that a discrete group is C*-simple if and only if it has Powers' averaging property. Second, we prove that a discrete grou…
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Locally Quasi-Convex Compatible Topologies on a Topological Group Open
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obvious…
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C*‐norms for tensor products of discrete group C*‐algebras Open
Let be a discrete group. We show that if is nonamenable, then the algebraic tensor products and do not admit unique ‐norms. Moreover, when and are discrete groups containing copies of noncommutative free groups, then and admit ‐norms. Anal…
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Stable homology of surface diffeomorphism groups made discrete Open
We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that [math] –diffeomorphisms of surfaces as family of discrete groups exhibit homological stabili…
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Convex cocompact actions of relatively hyperbolic groups Open
In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be r…
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Ergodic geometry for non-elementary rank one manifolds Open
Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ whichcontains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to…
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Property A and uniform embedding for locally compact groups Open
For locally compact groups, we define an analogue to Yu’s property A that he defined for discrete metric spaces.We show that our property A for locally compact groups agrees with Roe’s notion of property A for proper metric spaces, defined…
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Approximating simple locally compact groups by their dense locally compact subgroups Open
The class, denoted by $\mathscr{S}$, of totally disconnected locally compact groups which are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups…
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Infinite volume and infinite injectivity radius Open
We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group, and let $\Lambda\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $\Lambda\backslash G/K$ admits injected balls o…
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Reflection groups and discrete integrable systems Open
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Dis…