Displaying similar documents to “Counting arithmetic subgroups and subgroup growth of virtually free groups”

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let Γ be a group and r n ( Γ ) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function 𝒵 Γ ( s ) = n = 1 r n ( Γ ) n - s . When Γ is an arithmetic group satisfying the congruence subgroup property then 𝒵 Γ ( s ) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

A note on infinite a S -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an a S -group. We study some properties of a S -groups. For instance, it is shown that a nilpotent group G is an a S -group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an a S -group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group W and a Coxeter element c of W ; the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

An explicit computation of p -stabilized vectors

Michitaka MIYAUCHI, Takuya YAMAUCHI (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we give a concrete method to compute p -stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over p -adic fields. An application to the global setting is also discussed. In particular, we give an explicit p -stabilized form of a Saito-Kurokawa lift.

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

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A geometric progression of length k and integer ratio is a set of numbers of the form a , a r , . . . , a r k - 1 for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ( a i ) i = 1 of positive real numbers with a₁ = 1 such that the set G ( k ) = i = 1 ( a 2 i , a 2 i - 1 ] contains no geometric progression of length k and integer ratio. Moreover, G ( k ) is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...

A note on sumsets of subgroups in * p

Derrick Hart (2013)

Acta Arithmetica

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Let A be a multiplicative subgroup of * p . Define the k-fold sumset of A to be k A = x 1 + . . . + x k : x i A , 1 i k . We show that 6 A * p for | A | > p 11 / 23 + ϵ . In addition, we extend a result of Shkredov to show that | 2 A | | A | 8 / 5 - ϵ for | A | p 5 / 9 .

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field K with 3 -class group Cl 3 ( K ) of type ( 3 , 3 ) , the structure of the 3 -class groups Cl 3 ( N i ) of the four unramified cyclic cubic extension fields N i , 1 i 4 , of K is calculated with the aid of presentations for the metabelian Galois group G 3 2 ( K ) = Gal ( F 3 2 ( K ) | K ) of the second Hilbert 3 -class field F 3 2 ( K ) of K . In the case of a quadratic base field K = ( D ) it is shown that the structure of the 3 -class groups of the four S 3 -fields N 1 , ... , N 4 frequently determines the type of principalization of the 3 -class group of K in N 1 , ... , N 4 . This...

On the structural theory of  II 1 factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor L Γ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that L Γ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in  Sp ( n , 1 ) , n 2 , are virtually W * -superrigid.

On solvability of finite groups with some s s -supplemented subgroups

Jiakuan Lu, Yanyan Qiu (2015)

Czechoslovak Mathematical Journal

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A subgroup H of a finite group G is said to be s s -supplemented in G if there exists a subgroup K of G such that G = H K and H K is s -permutable in K . In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group G is solvable if every subgroup of odd prime order of G is s s -supplemented in G , and that G is solvable if and only if every Sylow subgroup of odd order of G is s s -supplemented in G . These results...

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let k be a p -adic field. Let G be the group of k -rational points of a connected reductive group 𝖦 defined over k , and let 𝔤 be its Lie algebra. Under certain hypotheses on 𝖦 and k , wethe tempered dual G ^ of G via the Plancherel formula on 𝔤 , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on 𝔤 and G . As a consequence, we prove that any tempered representation contains a good minimal 𝖪 -type; we extend this result to irreducible...

An arithmetic Riemann-Roch theorem for pointed stable curves

Gérard Freixas Montplet (2009)

Annales scientifiques de l'École Normale Supérieure

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Let ( 𝒪 , Σ , F ) be an arithmetic ring of Krull dimension at most 1, 𝒮 = Spec 𝒪 and ( π : 𝒳 𝒮 ; σ 1 , ... , σ n ) an n -pointed stable curve of genus g . Write 𝒰 = 𝒳 j σ j ( 𝒮 ) . The invertible sheaf ω 𝒳 / 𝒮 ( σ 1 + + σ n ) inherits a hermitian structure · hyp from the dual of the hyperbolic metric on the Riemann surface 𝒰 . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω 𝒳 / 𝒮 ( σ 1 + ... + σ n ) hyp . The theorem is applied to modular curves X ( Γ ) , Γ = Γ 0 ( p ) or Γ 1 ( p ) , p 11 prime, with sections given by the cusps. We show Z ' ( Y ( Γ ) , 1 ) e a π b Γ 2 ( 1 / 2 ) c L ( 0 , Γ ) , with p 11 m o d 12 when Γ = Γ 0 ( p ) . Here Z ( Y ( Γ ) , s ) is the Selberg...