Our first construction is very similar in spirit to an analogous one in Euclidean space. The group of isometries of Euclidean R is given by O(n) ⊕ R or SO(n) ⊕ R if we want to preserve orientation; the first factor is made up of rotations and reflections (in the non-orientation-preserving case) about the origin, while the second factor gives translations. Now, it turns out that O(n) is generate...

Hyperbolic fibrations of PG(3, q) were introduced by Baker, Dover, Ebert and Wantz in [1]. Since then, many examples were found, all of which are regular and agree on a line. It is known, via algebraic methods, that a regular hyperbolic fibration of PG(3, q) that agrees on a line gives rise to a flock of a quadratic cone in PG(3, q), and conversely. In this paper this correspondence will be exp...

For a triangle in the hyperbolic plane, let α, β, γ denote the angles opposite the sides a, b, c, respectively. Also, let h be the height of the altitude to side c. Under the assumption that α, β, γ can be chosen uniformly in the interval (0, π) and it is given that α + β + γ < π, we show that the strong triangle inequality a + b > c + h holds approximately 79% of the time. To accomplish this, ...

In this note, we present a short trigonometric proof to the Steiner Lehmus Theorem in hyperbolic geometry. 2000 Mathematics Subject Classification: 30F45, 20N99, 51B10, 51M10

In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles.

[1] L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. Math. 72 (1960), pp. 413– 429 [2] F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. Math. 124 (1986), pp. 71–158 [3] D. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, in Analytical and Geometric Aspects of Hyperbolic Space, LMS 111 (198...

This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space. The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal. The infant subject grew very rapidl...

Turn the set of permutations of n objects into a graph Gn by connecting two permutations that differ by one transposition, and let σt be the continuous time simple random walk on this graph. In a previous paper, Berestycki and Durrett (2004) showed that the limiting behavior of the distance from the identity at time cn/2 has a phase transition at c = 1. When c < 1, it is asymptotically cn/2, wh...

We consider central limit theorems and their generalizations for matrix groups acting co-compactly or convex co-compactly on the hyperbolic plane. We consider statistical results for the displacement in the hyperbolic metric, the action on the boundary and the relationship with classical matrix groups.

We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y2. Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected...