Estimates for the hyperbolic metric of the punctured plane and applications

The hyperbolic metrich ^Ω of the twice punctured complex plane Ω is studied. A new recursive algorithm for computing the density λ_Ω ofh ^Ω is given. For a proper subdomainG of Ω we answer a question of G. Martin concerning quasiconformal mappings ofG that can be extended to the complement ofG as the identity map.     

Eine Verallgemeinerung des Satzes von Holditch

The Holditch Theorems for the euclidian plane and the euclidian sphere are generalized for the euclidian space. If we move a hyperbolic linear congruence of linesN along a closed ruled surfaceX, each line ofN generates a closed ruled surfaceY. Some global invariants ofY are linear functions of some global invariants ofX.

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Herrn Prof. Dr. K. Strubecker zum 70. Geburtstag gewidmet     

Regular simplices and lower volume bounds for hyperbolicn-manifolds

For an-dimensional compact hyperbolic manifoldM ⁿ a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofM ⁿ. Its proof relies upon local density bounds for hyperbolic sphere packings.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

Onq-difference equations andZ n decompositions of exp q function

Theq-extended hyperbolic functions ofn-th order {h _q,s(z)}s∈ Z _n which areZ _n-components of expq function form the set fundamental solutions of a simpleq-difference equation. Against the background ofq-deformed hyperbolic functions ofn-th order other natural extensions and related topics are considered. Apart from easy general solution of homogenous ordinaryq-difference equations ofn-th order main trigonometric-like identity known for hyperbolic functions ofn-th order is given itsq-commutative counterpart. Hint how to arrive at other identities is implicit in theq-treatment proposed. The paper constitutes an example of the application of the method of projections presented in Advances in Applied Clifford Algebras publication [19]; see also references to Ben Cheikh’s papers.     

Hyperbolic unitals in the Hall planes

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.     

Properties of paradoxical sets in the plane

We show that every paradoxical subset of ² has empty interior, and every measurable paradoxical subset of ² has measure zero. We investigate how the proof fails in the hyperbolic plane, where there are paradoxical sets with interior points and with infinite measure.     

The Diameter of a Hyperbolic Disc

Recently, Matti Vuorinen asked whether the set-theoretic diameter of a hyperbolic disc of radius r in a hyperbolic plane region Ω is 2r. The answer is affirmative if Ω is simply or doubly connected. However, there are a hyperbolic discs in the triply-punctured sphere whose set-theoretic diameter is less than twice the radius. Also, for finitely connected hyperbolic plane regions all hyperbolic discs sufficiently close to the boundary have set-theoretic diameter equal to twice the radius. Precisely, if Ω is a hyperbolic plane region of finite connectivity, then there is a compact subset K of Ω such that any hyperbolic disc which is disjoint from K has diameter equal to twice the radius.     

Height in splittings of hyperbolic groups

SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of [23].     

Delaunay triangulation and the convex hull ofn points in expected linear time

An algorithm is presented which produces a Delaunay triangulation ofn points in the Euclidean plane in expected linear time. The expected execution time is achieved when the data are (not too far from) uniformly distributed. A modification of the algorithm discussed in the appendix treats most of the non-uniform distributions. The basis of this algorithm is a geographical partitioning of the plane into boxes by the well-known Radix-sort algorithm. This partitioning is also used as a basis for a linear time algorithm for finding the convex hull ofn points in the Euclidean plane.     

Hyperbolic lengths and conformal embeddings of Riemann surfaces

Leth be a homeomorphic bijection between hyperbolic Riemann surfacesR andR’. If there is a conformal mapping ofR intoR’ homotopic toh, then for any hyperbolic geodesicc onR the length of the hyperbolic geodesic freely homotopic to the imageh(c) is less than or equal to the hyperbolic length ofc. We show that the converse is not necessarily true.     

Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

A lattice point problem on the regular tree

Huber (1956) [8] considered the following problem on the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x∈H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree.     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In [10] one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane \mathbb E²{{\mathbb E}^2} were obtained. Moreover, the relations between the complex velocities of one-parameter motion in the complex plane were provided by [10]. One-parameter planar homothetic motion was defined in the complex plane, [9]. In this paper, analogous to homothetic motion in the complex plane given by [9], one-parameter planar homothetic motion is defined in the hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale h identically equal to 1, the results given in [15] are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.     

Hyperbolic euler formula in the n-dimensional Minkowski space

In this paper, the properties ofn-dimensional Minkowski space discussed by using Clifford algebra. The hyperbolic Euler formula is given inn-dimensional Minkowski space.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

On hyperbolic right-angled polygons

In this paper we find conditions in order to construct hyperbolic right-angledN-gons with the lengths ofN-3 sides given.Explicit formulae for the length of a side in terms of the lengths ofN-3 non-adjacent sides are obtained.Partially supported by CICYT.     

Coordinates in the hyperbolic plane

Abstract This note shows that in the hyperbolic plane three kinds of coordinates are possible. Keywords: Hyperbolic plane, Quasi-regular quadrangle, Pseudo-parallelogram, Hypercycle, Hypocycle Mathematics Subject Classification (2000): 20N05     

On the definition of baer subplanes of infinite planes

LetP be an infinite projective plane and letB be a subplane ofP. An example is given which shows that the following two conditions are independent: (1) Every line ofP is incident with at least one point ofB. (2) Every point ofP is incident with at least one line ofB.This research was supported by the C.N.R. (Italy).