scholarly journals Fundamental domains for genus-zero and genus-one congruence subgroups

2010 ◽  
Vol 13 ◽  
pp. 222-245
Author(s):  
C. J. Cummins
Keyword(s):  
Fundamental Domain ◽  
Congruence Subgroup ◽  
Discrete Subgroup ◽  
Magma Source ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Fundamental Domains ◽  
Finite Set ◽  
Data Files ◽  
Genus One

AbstractIn this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups ofPSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integerf, which determines a maximal discrete subgroup Γ0(f)+ofSL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+is in a subgroupG; and the index ofGin Γ0(f)+. The output consists of: a fundamental domain forG, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators ofG. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

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.

scholarly journals VIRASORO CORRELATION FUNCTIONS FOR VERTEX OPERATOR ALGEBRAS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250106 ◽  
Author(s):  
DONNY HURLEY ◽  
MICHAEL P. TUITE
Keyword(s):  
Graph Theory ◽  
Vertex Operator ◽  
Generating Functions ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Genus Zero ◽  
Genus One

We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.

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 $\mathbf{SL}_{n}(\mathbb{Z})$.

scholarly journals TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

2021 ◽  
pp. 1-62
Author(s):  
Zongbin Chen
Keyword(s):  
Fundamental Domain ◽  
Recurrence Relations ◽  
Rational Points ◽  
Orbital Integrals ◽  
Orbital Integral ◽  
Comparison Of Results ◽  
Fundamental Domains ◽  
Springer Fiber ◽  
Springer Fibers

Abstract We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets

2018 ◽  
Vol 9 (2) ◽  
pp. 93-107
Author(s):  
Heidi Burgiel ◽  
Vignon Oussa
Keyword(s):  
Fundamental Domain ◽  
Full Rank ◽  
Unit Cube ◽  
Gabor Frames ◽  
Invertible Matrix ◽  
Orthonormal Bases ◽  
Fundamental Domains ◽  
Uncountable Family ◽  
Indicator Functions

AbstractThe main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in{\mathbb{R}^{d}}admitting common connected fundamental domains of the type{N[0,1)^{d}}, whereNis an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type{N[0,1)^{d}}. We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type{N[0,1)^{2}}, whereNis an invertible matrix.

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 CONVEX STANDARD FUNDAMENTAL DOMAIN FOR SUBGROUPS OF HECKE GROUPS

2010 ◽  
Vol 83 (1) ◽  
pp. 96-107
Author(s):  
BOUBAKARI IBRAHIMOU ◽  
OMER YAYENIE
Keyword(s):  
Fundamental Domain ◽  
Discrete Subgroup ◽  
Practical Method ◽  
Locally Finite ◽  
Hecke Groups ◽  
Modular Groups ◽  
Group Form ◽  
Image Position ◽  
The Right ◽  
Hyperbolic Polygon

AbstractIt is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

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