The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every nonsingular, prime alternating link. These tilings define a combinatorial space at infinity, similar to the space at infinity for word hyperbolic groups.

Jørgensen’s inequality gives a necessary condition for a non-elementary two generator group of isometries of real hyperbolic 2-space to be discrete. We give analogues of Jørgensen’s inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Mathematics Subject Classifications (2000): 20H10, 22E40, 57S30.

The theme of this survey is that subgroups of the mapping class group MCG of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of Teichmüller space T = T ∪PMF , just as discrete subgroups of the isometries of hyperbolic space Hn can be studied via their action on compactified hyperbolic space H n = Hn ∪ S ∞ . 2000 Math...

Multistep methods for firstand second-order ordinary differential equations are used for the full discretizations of standard Galerkin approximations to the initial-periodic boundary value problem for first-order linear hyperbolic equations in one space dimension and to the initial-boundary value problem for second-order lin2 ear selfadjoint hyperbolic equations in many space dimensions. L -err...

Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space. We explore this representation using the isometric action of the group of Möbius transformations on hyperbolic space, and prove a classical theorem on continued fractions.

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the innnite curvature limit of the space of hyperbolic monopoles and show that the associated rational maps appear explicitly here.

Complex hyperbolic triangle groups are representations of a hyperbolic (p, q, r) reflection triangle group to the group of holomorphic isometries of complex hyperbolic space H C , where the generators fix complex lines. In this paper, we obtain all the discrete and faithful complex hyperbolic (3, 3, n) triangle groups. Our result solves a conjecture of Schwartz [16] in the case when p = q = 3.