# Realising end invariants by limits of minimally parabolic, geometrically finite groups

```@article{Ohshika2005RealisingEI,
title={Realising end invariants by limits of minimally parabolic, geometrically finite groups},
author={Ken'ichi Ohshika},
journal={Geometry \& Topology},
year={2005},
volume={15},
pages={827-890}
}```
We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol…
41 Citations
THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II: THE ENDING LAMINATION CONJECTURE
• Mathematics
• 2004
Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold N with nitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this
Non-realizability and ending laminations: Proof of the density conjecture
• Mathematics
• 2012
We give a complete proof of the Bers–Sullivan–Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is
Constructing knots with specified geometric limits
• Mathematics
• 2022
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
Deformation Spaces of Kleinian Surface Groups are not Locally Connected.
For any closed surface S of genus g 2, we show that the deformation space AH.S I/ of marked hyperbolic 3‐manifolds homotopy equivalent to S is not locally connected. This proves a conjecture of
Complex and quaternionic hyperbolic Kleinian groups with real trace fields
• Mathematics
J. Lond. Math. Soc.
• 2016
IfGamma is irreducible, it is shown that if the trace field of \$\Gamma\$ is contained in R, it preserves a totally geodesic submanifold of constant negative sectional curvature.
Effective drilling and filling of tame hyperbolic 3-manifolds
• Mathematics
• 2021
. We give eﬀective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn ﬁllings. In the thin parts of the manifold, we give eﬀective
Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces
In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit
On the number of connected components of the set of fixed points of isometries of geometrically finite hyperbolic manifolds
• Mathematics
• 2017
Let \$M\$ be a geometrically finite \$m\$-dimensional hyperbolic manifold, where \$m \geq 2\$, and let \$G\$ be a finite group of its isometries. We provide a formula for the number of connected components
Dynamics on PSL(2, C)-Character Varieties of Certain Hyperbolic 3-Manifolds.
Let M be a compact orientable hyperbolizable 3-manifold. In this thesis we study the action of the group of outer automorphisms of the fundamental group of M on the PSL(2,C)-character variety of M.
Local topology in deformation spaces of hyperbolic 3-manifolds
• Mathematics
• 2011
We prove that the deformation space AH.M/ of marked hyperbolic 3‐manifolds homotopy equivalent to a fixed compact 3‐manifold M with incompressible boundary is locally connected at minimally parabolic

## References

SHOWING 1-10 OF 103 REFERENCES
Geometrically finite kleinian groups and parabolic elements
Let r be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of P can be simultaneously made parabolic in a group on the
THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II: THE ENDING LAMINATION CONJECTURE
• Mathematics
• 2004
Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold N with nitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this
The classification of punctured-torus groups.
Thurston’s ending lamination conjecture proposes that a flnitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that
Convergence of freely decomposable Kleinian groups
• Mathematics
• 2007
We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary
Kleinian groups and the complex of curves
We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our
Uniform perfectness of the limit sets of Kleinian groups
In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of
Surface groups and 3-manifolds which fiber over the circle
Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose
The classification of Kleinian surface groups I : Models and bounds : preprint
We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model” for the
Rigidity and topological conjugates of topologically tame Kleinian groups
Minsky proved that two Kleinian groups G1 and G2 are quasiconformally conjugate if they are freely indecomposable, the injectivity radii at all points of H/G1, H/G2 are bounded below by a positive