Hyperbolic unit groups

Hyperbolic Unit Groups and Quaternion Algebras

We classify the quadratic extensions K = Q[ √ d] and the finite groups G for which the group ring oK [G] of G over the ring oK of integers of K has the property that the group U1(oK [G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(oK) of the quaternion algebra H(K) = ` −1, −1 K ́ , when it is a division algebra. Mathematics Subject Classification. Primary [...

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

Lacunary hyperbolic groups

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptot...

Hyperbolic Groups Lecture Notes

Complex Hyperbolic Triangle Groups

The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simplest nontrivial complex hyperbolic discrete groups. In particular, I will talk...