Topologically conjugate Kleinian groups

Two Kleinian groups and are said to be topologically conjugate when there is a homeomorphism such that . It is conjectured that if two Kleinian groups and are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when is finitely generated and freely indecomposable, and the injectivity radii of all points of and are bounded below by a positive constant.

     

Endomorphisms of Kleinian groups

((without abstract)) Submitted: March 2001; Revision: October 2002; Final version: January 2003     

Strong convergence of Kleinian groups

In this paper, we consider algebraically convergent sequences of quasi-conformal deformations of a convex cocompact Kleinian group. We characterize those sequences which converge strongly in terms of their Ahlfors–Bers parameterizations.Received: December 2003 Revision: November 2004 Accepted: November 2004     

Incommensurability criteria for Kleinian groups

The purpose of this note is to present a criterion for an infinite collection of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes.) Namely, such a collection is commensurably infinite if there is a uniform upper bound on the volumes of the manifolds in .

There is a related criterion for an infinite collection of distinct finitely generated Kleinian groups with non-empty domain of discontinuity to be commensurably infinite. (Here, a collection of Kleinian groups is commensurably infinite if it is infinite modulo the combined equivalence relations of commensurability and conjugacy in .) Namely, such a collection is commensurably infinite if there is a uniform bound on the areas of the quotient surfaces of the groups in .

     

Bounded geometry for Kleinian groups

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Oblatum 31-VII-2000 & 9-V-2001?Published online: 20 July 2001     

Triangle subgroups of Kleinian groups

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF ²=∏/Γ _∏ is always embedded in the hyperbolic 3-orbifoldM ³=ℍ³/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup. We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups. for J. A. Kalman on the occasion of his 65th birthday Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.     

Families of univalent functions and Kleinian groups

The paper treats two topics. We obtain new characterizations of quasidiscs using theλ-lemma. We study regularb-groups by combining results on holomorphic families of injections with theorems on algebraic convergence of Kleinian groups and the structure of the Eichler cohomology spaces. We also obtain results about projective structures whose developing maps are coverings. Research partially supported by NSF grant DMS 8401280.