Normal Congruence Subgroups of Hecke Groups

2008 ◽  
Vol 15 (04) ◽  
pp. 707-720
Author(s):  
Bo Chen ◽  
Pingzhi Yuan
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Finite Index ◽  
Principal Congruence ◽  
Important Class ◽  
Congruence Subgroups ◽  
Discrete Subgroups ◽  
Hecke Groups ◽  
Hecke Group ◽  
Normal Congruence

Hecke groups are an important class of discrete subgroups of PSL(2, ℝ), which play an important role in the study of Dirichlet series. Subgroups with finite index of a Hecke group, which are called congruence subgroups, are often used. Let q be a positive integer with [Formula: see text]. For the Hecke group [Formula: see text], the structures of principal congruence subgroups and normal congruence subgroups of level m are investigated in many papers, where m is a prime or a power of an odd prime. In this paper, we deal with the case that the level m is a power of 2.

scholarly journals Principal Congruence Subgroups of the Hecke Groups

2000 ◽  
Vol 85 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Mong-Lung Lang ◽  
Chong-Hai Lim ◽  
Ser-Peow Tan
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups

Principal congruence subgroups of the Hecke groups and related results

2009 ◽  
Vol 40 (4) ◽  
pp. 479-494 ◽  
Author(s):  
Sebahattin Ikikardes ◽  
Recep Sahin ◽  
I. Naci Cangul
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups

scholarly journals Regular maps and principal congruence subgroups of Hecke groups

2005 ◽  
Vol 26 (3-4) ◽  
pp. 437-456 ◽  
Author(s):  
Ioannis Ivrissimtzis ◽  
David Singerman
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups ◽  
Regular Maps

Power and free normal subgroups of generalized Hecke groups

2019 ◽  
Vol 13 (04) ◽  
pp. 2050080
Author(s):  
Recep Sahin ◽  
Taner Meral ◽  
Özden Koruoğlu
Keyword(s):  
Positive Integer ◽  
Normal Subgroups ◽  
Hecke Groups ◽  
Hecke Group

Let [Formula: see text] and [Formula: see text] be integers such that [Formula: see text] [Formula: see text] and let [Formula: see text] be generalized Hecke group associated to [Formula: see text] and [Formula: see text] Generalized Hecke group [Formula: see text] is generated by [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] In this paper, for positive integer [Formula: see text] we study the power subgroups [Formula: see text] of generalized Hecke groups [Formula: see text]. Also, we give some results about free normal subgroups of generalized Hecke groups [Formula: see text]

Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

On Certain Finitely Generated Subgroups of Groups Which Split

2003 ◽  
Vol 46 (1) ◽  
pp. 122-129 ◽  
Author(s):  
Myoungho Moon
Keyword(s):  
Positive Integer ◽  
Free Product ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Product With Amalgamation

AbstractDefine a group G to be in the class 𝒮 if for any finitely generated subgroup K of G having the property that there is a positive integer n such that gn ∈ K for all g ∈ G, K has finite index in G. We show that a free product with amalgamation A *CB and an HNN group A *C belong to 𝒮, if C is in 𝒮 and every subgroup of C is finitely generated.

APPROXIMATION OF -ANALYTIC TORSION FOR ARITHMETIC QUOTIENTS OF THE SYMMETRIC SPACE

2018 ◽  
Vol 19 (2) ◽  
pp. 307-350
Author(s):  
Jasmin Matz ◽  
Werner Müller
Keyword(s):  
Symmetric Space ◽  
Principal Congruence ◽  
Analytic Torsion ◽  
Congruence Subgroups ◽  
Arithmetic Subgroup ◽  
Limiting Behavior

In [31] we defined a regularized analytic torsion for quotients of the symmetric space $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^{2}$-analytic torsion.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

scholarly journals SPECIAL VALUES OF DIRICHLET SERIES AND ZETA INTEGRALS

2012 ◽  
Vol 08 (03) ◽  
pp. 697-714 ◽  
Author(s):  
EDUARDO FRIEDMAN ◽  
ALDO PEREIRA
Keyword(s):  
Positive Integer ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Direct Calculation ◽  
Simple Relation ◽  
Product Rule ◽  
Special Values ◽  
Special Value

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = -N, obtained by analytic continuation of the Dirichlet series [Formula: see text], to special values of zeta integrals Z(s;f,g) = ∫x∊[0, ∞)p g(x)f(x)-s dx ( Re (s) ≫ 0). We prove a simple relation between ζ(-N;f,g) and Z(-N;fa, ga), where for a ∈ ℂp, fa(x) is the shifted polynomial fa(x) = f(a + x). By direct calculation we prove the product rule for zeta integrals at s = 0, degree (fh) ⋅ Z(0;fh, g) = degree (f) ⋅ Z(0;f, g) + degree (h) ⋅ Z(0;h, g), and deduce the corresponding rule for Dirichlet series at s = 0, degree (fh) ⋅ ζ(0;fh, g) = degree (f) ⋅ ζ(0;f, g)+ degree (h)⋅ζ(0;h, g). This last formula generalizes work of Shintani and Chen–Eie.

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