Spectral Estimates for Towers of Noncompact Quotients

1999 ◽  
Vol 51 (2) ◽  
pp. 266-293 ◽  
Author(s):  
Anton Deitmar ◽  
Werner Hoffman
Keyword(s):  
Analogous Result ◽  
Congruence Subgroup ◽  
Kernel Method ◽  
Linear Algebraic Group ◽  
Fixed Number ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Spectral Estimates ◽  
The Family ◽  
Upper Estimate

AbstractWe prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ.

scholarly journals On Conjugacy Classes of Congruence Subgroups of PSL(2, R)

2009 ◽  
Vol 12 ◽  
pp. 264-274 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Congruence Subgroup ◽  
Principal Congruence ◽  
Conjugacy Classes ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup ◽  
Genus One

AbstractLet G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup /overline Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

Free quotients of congruence subgroups of SL2 over a Dedekind ring of arithmetic type contained in a function field

1987 ◽  
Vol 101 (3) ◽  
pp. 421-429 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Function Field ◽  
Congruence Subgroup ◽  
Principal Congruence ◽  
Dedekind Ring ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup

Let R be a commutative ring with identity and let q be an ideal in R. For each n ≽ 2, let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. We put SLn(R, q) = Ker(SLn(R)→SLn(R/q)), the principal congruence subgroup of GLn(R) of level q. (By definition En(R, R) = En(R) and SLn(R, R) = SLn(R).)

scholarly journals The level four braid group

2018 ◽  
Vol 2018 (735) ◽  
pp. 249-264 ◽  
Author(s):  
Tara E. Brendle ◽  
Dan Margalit
Keyword(s):  
Braid Group ◽  
Symplectic Group ◽  
Congruence Subgroup ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Burau Representation ◽  
Dehn Twists

AbstractBy evaluating the Burau representation att=-1, one obtains a symplectic representation of the braid group. We study the resulting congruence subgroups of the braid group, namely, the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup is equal to the group generated by squares of Dehn twists. We also show that the image of the Brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup.

scholarly journals Modular symmetry at level 6 and a new route towards finite modular groups

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Cai-Chang Li ◽  
Xiang-Gan Liu ◽  
Gui-Jun Ding
Keyword(s):  
Invariant Theory ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Modular Invariant ◽  
Benchmark Model ◽  
Modular Groups ◽  
Principal Congruence Subgroup ◽  
Comprehensive Study

Abstract We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

scholarly journals Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

2020 ◽  
Vol 55 (4) ◽  
pp. 391-405
Author(s):  
E. A. Rovba ◽  
V. Yu. Medvedeva
Keyword(s):  
Rational Functions ◽  
Rational Interpolation ◽  
Asymptotic Equality ◽  
Fixed Number ◽  
Theoretical Research ◽  
Extended System ◽  
Uniform Approximations ◽  
Upper Estimate ◽  
Interpolation Functions ◽  
Interpolation Nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods. 

scholarly journals Torsion classes in the cohomology of congruence subgroups

1989 ◽  
Vol 105 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Dominique Arlettaz
Keyword(s):  
Prime Number ◽  
Congruence Subgroup ◽  
Surjective Homomorphism ◽  
Congruence Subgroups ◽  
Image Position ◽  
Homology Stability

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We defineusing upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

scholarly journals Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

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.

Sign in / Sign up

Export Citation Format

Share Document