Some remarks on the theorem of Möbius

H.S.M. Coxeter in his book Introduction to Geometry quotes a theorem of Möbius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.     

The Extension of Möbius Groups

Let G be a non-elementary subgroup of SL(2,Г _n ) containing hyperbolic elements. We show that G is the extension of a subgroup of SL(2,C) if and only if that G is conjugate in SL(2,Г _n ) to a group G' with the following properties: (1) There are g ₀, h ? G', where g ₀ and h are hyperbolic, such that fix(g ₀) = {0,∞}, fix(h)∩fix(g ₀) =  and fix(h) ∩ C ≠ ; (2) tr(g) ? C for each g ? G'. As an application, we show that if G contains only hyperbolic elements and uniformly parabolic elements, then G is the extension of a subgroup of SL(2,C), which also yields the discreteness of G.     

The Möbius function on Abelian semigroups

Let X be an Abelian semigroup such that the following conditions hold: (i) if x × y = II (II is the identity element), then x = y = II; (ii) the set {{x, y}: x × y = a} is finite for any a ∈ X. Let Λ be any field, and let ℰ be the algebra of all Λ-valued functions on X. The convolution of u, υ ∈ ℰ is defined by

Harmonic Analysis on the Möbius Gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball \({{\mathbb {B}}_{t}^{n}}\), a model of the \(n\)-dimensional real hyperbolic space. The Poincaré ball \({{\mathbb {B}}_{t}^{n}}\) is the open ball of the Euclidean \(n\)-space \(\mathbb {R}^n\) with radius \(t >0\), centered at the origin of \(\mathbb {R}^n\) and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in \(\mathbb {R}^n.\) For any \(t>0\) and an arbitrary parameter \(\sigma \in \mathbb {R}\) we study the \((\sigma ,t)\)-translation, the \((\sigma ,t)\)-convolution, the eigenfunctions of the \((\sigma ,t)\)-Laplace–Beltrami operator, the \((\sigma ,t)\)-Helgason Fourier transform, its inverse transform and the associated Plancherel’s Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when \(t \rightarrow +\infty \) the resulting hyperbolic harmonic analysis on \({{\mathbb {B}}_{t}^{n}}\) tends to the standard Euclidean harmonic analysis on \(\mathbb {R}^n,\) thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on \({{\mathbb {B}}_{t}^{n}}\).     

The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into ? ⁿ , yields no information on the quasiconformality or quasisymmetry of a topological embedding of ? ^k into ? ⁿ for 1 < k ≤ n. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” MMC(f) ≤ H < 1. We prove that if this condition is fulfilled then every homeomorphism of domains in \(\overline {\mathbb{R}^n }\) is K(H)-quasiconformal, while a topological embedding of the sphere \(\overline {\mathbb{R}^k }\) into \(\overline {\mathbb{R}^n }\) (for 1 ≤ k ≤ n) is ω _H-quasimöbius. The quasiconformality coefficient of K(H) and the distortion function ω _H depend only on H and are expressed by explicit formulas showing that K(H) → 1 and ω _H → id as H → 1/2. Since MMC(f) = 1/2 is equivalent to the Möbius property of f, the resulting formulas yield the closeness of the mapping to a Möbius mapping for H near 1/2.     

On the Möbius Function of a Pointed Graded Lattice

In this paper, we compute the Möbius function of pointed integer partition and pointed ordered set partition using topological and analytic methods. We show that the associated order complex is a wedge of spheres and compute the associated reduced homology group for each subposet. In addition, we compute the Möbius function of pointed graded lattice and use our method to compute the Möbius function of pointed direct sum decomposition of vector spaces.     

Möbius bilipschitz homogeneous arcs on the plane

A möbius bilipschitz mapping is an η-quasimöbius mapping with the linear distortion function η(t) = Kt. We show that if an open Jordan arc γ ? C with distinct endpoints a and b is homogeneous with respect to the family F_K of möbius bilipschitz automorphisms of the sphere C with K specified then γ has bounded turning RT(γ) in the sense of Rickman and, consequently, γ is a quasiconformal image of a rectilinear segment. The homogeneity of γ with respect to F_K means that for all x, y ∈ γ {a, b} there exists f ∈ F_K with f(γ) = γ and f(x) = y. In order to estimate RT(γ) from above, we introduce the condition BR(δ) of bounded rotation of γ, and then the explicit bound depends only on K and δ.     

Birkhoff interpolation on unity and on Möbius transform of the roots of unity

In this paper the regularity of two interpolation problems is proved. One uses (0,1,..., r-2,r) interpolation with as nodes Möbius transforms of nth roots of unity with the point z=1 added and the other is concerned with Pál-type interpolation on the pair of polynomials (z+α)ⁿ+(1+α z)ⁿ and (z+α)ⁿ-(1+α z)ⁿ, where 0 < α < 1.     

On partial sums of a spectral analogue of the Möbius function

Sankaranarayanan and Sengupta introduced the function μ ^*(n) corresponding to the Möbius function. This is defined by the coefficients of the Dirichlet series 1/L _f (s), where L _f (s) denotes the L-function attached to an even Maaß cusp form f. We will examine partial sums of μ ^*(n). The main result is $\sum_{n\leq x}\mu^{*}(n)=O(x\exp(-A\sqrt{\log x}))$ , where A is a positive constant. It seems to be the corresponding prime number theorem.     

The r-Stirling numbers of the first kind in terms of the Möbius function

The Ramanujan Journal - The main result of this paper is an identity expressing the r-Stirling number of the first kind as a sum involving binomial coefficients and the Möbius function of the...     

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.     

Pairwise products of moduli of families of curves on a Riemannian Möbius strip

We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus and extremal metric).     

The Möbius Category of Some Combinatorial Inverse Semigroups

\noindent The purpose of this paper is to point out some aspects of the relationship between combinatorial inverse semigroups and their Möbius categories, and to explore combinatorial results arising from combinatorial Brandt semigroups, fundamental simple inverse -semigroups and from the free monogenic inverse semigroup.     

Metric Möbius geometry and a characterization of spheres

We obtain a Möbius characterization of the n-dimensional spheres S ⁿ endowed with the chordal metric d ₀. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S ⁿ , d ₀) for some n ≥ 1.     

A four-point criterion for the Möbius property of a homeomorphism of plane domains

An ordered quadruple of pairwise distinct points T = {z ₁, z ₂, z ₃, z ₄} ? C is called regular whenever z ₂ and z ₄ lie at the opposite sides of the line through z ₁ and z ₃. Consider Φ(T) = ∠z ₁ z ₂ z ₃ + ∠z ₁ z ₄ z ₃ (the angles are undirected) as some geometric characteristic of a regular tetrad. We prove the following theorem: For every fixed α ∈ (0, 2π) the Möbius property of a homeomorphism f: D → D* of domains in C is equivalent to the requirement that each regular tetrad T ? D with Φ(T) = α whose image fT is also a regular tetrad satisfies Φ(fT) = α. In 1994 Haruki and Rassias established this criterion for the Möbius property only in the class of univalent analytic functions f(z).     

On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformations

Summary The main result of this paper is the determination of maximal solutions of the equation u =u ^{(n + 2)/(n – 2)} arising from the conformal change of a flat metric in a domain (in euclidean space) to a metric of negative scalar curvature. These solutions are obtained on the upper half ball (and therefore on anything conformally equivalent to it). There are a couple of applications using basic comparison principle arguments. Namely, the associated metric on a domain induced by a maximal solution is shown to be comparable to the quasihyperbolic metric for nice domains. Also an analogue of the isoperimetric inequality is found for the harmonic radius.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Maximizing the Möbius Function of a Poset and the Sum of the Betti Numbers of the Order Complex

Received: February 28, 1995/Revised: Revised October 30, 1998