scholarly journals Geometry of Neumann subgroups

1989 ◽  
Vol 47 (3) ◽  
pp. 350-367 ◽  
Author(s):  
Ravi S. Kulkarni
Keyword(s):  
Parabolic Subgroup ◽  
Fuchsian Group ◽  
Hyperbolic Plane ◽  
Modular Group ◽  
Natural Extension ◽  
Geometric Method ◽  
General Context ◽  
Open Disk ◽  
Finitely Generated ◽  
Maximal Parabolic Subgroup

AbstractA Neumann subgroup of the classical modular group is by definition a complement of a maximal parabolic subgroup. Recently Neumann subgroups have been studied in a series of papers by Brenner and Lyndon. There is a natural extension of the notion of a Neumann subgroup in the context of any finitely generated Fuchsian group Γ acting on the hyperbolic plane H such that Γ/H is homeomorphic to an open disk. Using a new geometric method we extend the work of Brenner and Lyndon in this more general context.

scholarly journals Diophantine approximation on Hecke groups

1985 ◽  
Vol 27 ◽  
pp. 117-127 ◽  
Author(s):  
J. Lehner
Keyword(s):  
Fuchsian Group ◽  
Modular Group ◽  
Irrational Number ◽  
Upper And Lower Bounds ◽  
Limit Set ◽  
Finitely Generated ◽  
Best Constant ◽  
Hecke Groups ◽  
Image Position ◽  
Hurwitz Constant

If α is a real irrational number, there exist infinitely many reduced rational fractions p/q for whichand √5 is the best constant possible. This result is due to A. Hurwitz. The following generalization was proposed in [2]. Let Г be a finitely generated fuchsian group acting on H+, the upper half of the complex plane. Let ℒ be the limit set of Г P and the set of cusps (parabolic vertices). Assume ∞∊P. Then if α∊ℒ–P, we havefor infinitely many p/q∊Г(∞), where k depends only on Г. Attention centers onk running over the set for which (1.2) holds. We call hthe Hurwitz constant for Г. When Г=Г(1), the modular group, (1.2) reduces to (1.1) and h(Г(l))=√5. A proof of (1.2) when Г is horocyclic (i.e., ℒ=ℝ, the real axis) was furnished by Rankin [4]; he also found upper and lower bounds for h. See also [3, pp. 334–5], where the theorem is proved for arbitrary finitely generated Г.

scholarly journals COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

2018 ◽  
Vol 98 (3) ◽  
pp. 422-433
Author(s):  
BORIS GOLDFARB ◽  
JONATHAN L. GROSSMAN
Keyword(s):  
Metric Spaces ◽  
General Context ◽  
Metric Geometry ◽  
Finitely Generated ◽  
Coherence Property ◽  
Finitely Generated Groups ◽  
Permanence Property ◽  
Permanence Properties ◽  
New Framework

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Convex fundamental domains of Fuchsian groups

1974 ◽  
Vol 76 (3) ◽  
pp. 511-513 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Fuchsian Group ◽  
Discrete Group ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Möbius Transformations ◽  
Fundamental Domains ◽  
Mobius Transformations

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.

On Representation of Integers from Thin Subgroups of SL(2, ℤ) with Parabolics

2018 ◽  
Vol 2020 (18) ◽  
pp. 5611-5629 ◽  
Author(s):  
Xin Zhang
Keyword(s):  
Critical Exponent ◽  
Lattice Point ◽  
Fuchsian Group ◽  
Spectral Gap ◽  
Circle Method ◽  
Inner Product ◽  
Point Counting ◽  
Finitely Generated ◽  
Exceptional Set ◽  
Representation Of Integers

Abstract Let $\Lambda <SL(2,\mathbb{Z})$ be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and $\mathbf{v},\mathbf{w}$ be two primitive vectors in $\mathbb{Z}^2\!-\!\mathbf{0}$. We consider the set $\mathcal{S}\!=\!\{\left \langle \mathbf{v}\gamma ,\mathbf{w}\right \rangle _{\mathbb{R}^2}\!:\!\gamma\! \in\! \Lambda \}$, where $\left \langle \cdot ,\cdot \right \rangle _{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy–Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich, and Sarnak, together with Gamburd’s 5/6 spectral gap, we show that if $\Lambda $ has parabolic elements, and the critical exponent $\delta $ of $\Lambda $ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain–Kontorovich, which proves a density-one statement for the case when $\Lambda $ is free, finitely generated, has no parabolics, and has critical exponent $\delta>0.999950$.

scholarly journals Higher-rank zeta functions andSLn-zeta functions for curves

2020 ◽  
Vol 117 (12) ◽  
pp. 6398-6408
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Zeta Function ◽  
Parabolic Subgroup ◽  
Vector Bundles ◽  
Reductive Group ◽  
Zeta Functions ◽  
Smooth Projective Curve ◽  
Maximal Parabolic Subgroup ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

scholarly journals Branching laws for small unitary representations of GL(n, ℂ)

2014 ◽  
Vol 25 (06) ◽  
pp. 1450052
Author(s):  
Jan Möllers ◽  
Benjamin Schwarz
Keyword(s):  
Parabolic Subgroup ◽  
Principal Series ◽  
Unitary Representations ◽  
Symmetric Pair ◽  
Maximal Parabolic Subgroup ◽  
Infinite Dimensional ◽  
Branching Laws ◽  
Series Representations ◽  
Unitary Principal Series ◽  
Kirillov Dimension

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

Transitivity for the modular group

1980 ◽  
Vol 88 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Mark Sheingorn
Keyword(s):  
Real Number ◽  
Half Plane ◽  
Limit Point ◽  
Fuchsian Group ◽  
Modular Group ◽  
Fundamental Region ◽  
Hyperbolic Line ◽  
Upper Half Plane

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.

scholarly journals Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Guangming Hu ◽  
Yutong Liu ◽  
Yu Sun ◽  
Xinjie Qian
Keyword(s):  
Harmonic Function ◽  
Unit Disk ◽  
Fuchsian Group ◽  
Fundamental Domain ◽  
Carleson Measure ◽  
Schwarzian Derivative ◽  
Finitely Generated ◽  
Schwarzian Derivatives ◽  
Carleson Condition

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

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