Related papers: Cores of hyperbolic 3-manifolds and limits of Klei…

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1…

Inspired by questions of convergence in continued fraction theory, Erd\H{o}s, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of M\"obius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with…

We give an expository account of our proof that each cusp-free hyperbolic 3-manifold M with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite hyperbolic 3-manifolds.

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup \Omega_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured…

We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we…

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the…

We study a generalization of the Fuchsian triangle groups to the hyperbolic 3-space, namely, the groups generated by half-turns in three hyperbolic lines. The role of the hyperbolic triangles is now played by the right-angled hexagons. This…

In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic $3$-space. In particular we make a detailed study of two-generator groups…

One of the basic problems in studying topological structures of deformation spaces for Kleinian groups is to find a criterion to distinguish convergent sequences from divergent sequences. In this paper, we shall give a sufficient condition…

In this article, we show that there exist discrete isometry groups of the $2$- and $3$-dimensional complex hyperbolic spaces with critical exponents arbitrarily close to but strictly smaller than the maximum possible value. This result…

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…

We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of…

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with…

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In…

These revised lecture notes are an expository account of part of the proof of Thurston's Ending Lamination Conjecture for Kleinian surface groups, which states that such groups are uniquely determined by invariants that describe the…

In this paper,based on the available mathematical works on geometry and topology of hyperbolic manifolds and discrete groups, some results of Freedman et al (hep-th/9804058) are reproduced and broadly generalized. Among many new results the…

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given…

We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of…

We consider the relation between geometrically finite groups and their limit sets in infinite-dimensional hyperbolic space. Specifically, we show that a rigidity theorem of Susskind and Swarup ('92) generalizes to infinite dimensions, while…