Surface subgroups of Kleinian groups with torsion

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.     

Endomorphisms of Kleinian groups

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

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.

     

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     

Geometric exponents and Kleinian groups

Suppose is the limit set of an analytically finite Kleinian group and that is an enumeration of the components of . Then This had been conjectured by Maskit. We also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincaré exponent. Oblatum 30-X-1995 & 11-III-1996