Hyperbolic angle

Abstract: In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

Abstract: If a hyperbolic link has a prime alternating diagram $D$, then we show that the link complement's volume can be estimated directly from $D$. We define a very elementary invariant of the diagram $D$, its twist number $t(D)$, and show that the volume lies between $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, where $v_3$ is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology.

Abstract: (1995). The Hyperbolic Number Plane. The College Mathematics Journal: Vol. 26, No. 4, pp. 268-280.

Abstract: Preface. 1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds. 2. Hyperbolic Sets of Diffeomorphisms. 3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets. 4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms. 5. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism. 6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and asymptotic Phase. 7. Hyperbolic Sets of Ordinary Differential Equations. 8. Transversal Homoclinic Points and Hyperbolic Sets in Differential Equations. 9. Shadowing Theorems for Hyperbolic Sets of Differential Equations. 10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations. 11. Numerical Shadowing. References.