# Hyperbolic limits of cantor set complements in the sphere

@article{Cremaschi2022HyperbolicLO, title={Hyperbolic limits of cantor set complements in the sphere}, author={Tommaso Cremaschi and Franco Vargas Pallete}, journal={Bulletin of the London Mathematical Society}, year={2022}, volume={54} }

Let M$M$ be a hyperbolic 3‐manifold with no rank two cusps admitting an embedding in S3$\mathbb {S}^3$ . Then, if M$M$ admits an exhaustion by π1$\pi _1$ ‐injective sub‐manifolds there exists Cantor sets Cn⊆S3$C_n\subseteq \mathbb {S}^3$ such that Nn=S3∖Cn$N_n=\mathbb {S}^3\setminus C_n$ is hyperbolic and Nn→M$N_n\rightarrow M$ geometrically.

## One Citation

Effective contraction of skinning maps

- Mathematics
- 2021

Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.

## References

SHOWING 1-10 OF 49 REFERENCES

Hyperbolization of infinite-type 3-manifolds

- Mathematics
- 2019

We study the class $\mathcal M^B$ of 3-manifolds $M$ that have a compact exhaustion $M=\cup_{i\in\mathbb N} M_i$ satisfying: each $M_i$ is hyperbolizable with incompressible boundary and each…

Algebraic and topological properties of big mapping class groups

- MathematicsAlgebraic & Geometric Topology
- 2018

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the…

Ends of hyperbolic 3-manifolds

- Mathematics
- 1993

Let N = H3/F be a hyperbolic 3-manifold which is homeomorphic to the interior of a compact 3-manifold. We prove that N is geometrically tame. As a consequence, we prove that F's limit set L. is…

Geometric limits of knot complements

- Mathematics
- 2010

We prove that any complete hyperbolic 3‐manifold with finitely generated fundamental group, with a single topological end, and which embeds into S3 is the geometric limit of a sequence of hyperbolic…

Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group

- Mathematics
- 2018

These notes are the English version of the paper "Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire".
The mapping class group Gamma of the complement of a Cantor…

Tameness of hyperbolic 3-manifolds

- Mathematics
- 2004

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved…

Existence of ruled wrappings in hyperbolic 3-manifolds

- Mathematics
- 2006

We present a short elementary proof of an existence theorem of certain CAT. 1/‐ surfaces in open hyperbolic 3‐manifolds. The main construction lemma in Calegari and Gabai’s proof of Marden’s Tameness…

On 3-manifolds

- Mathematics
- 2005

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P…

A locally hyperbolic 3-manifold that is not homotopy equivalent to any hyperbolic 3-manifold

- Mathematics
- 2018

We construct a locally hyperbolic 3-manifold $M$ such that $\pi_ 1(M)$ has no divisible subgroups. We then show that $M$ is not homotopy equivalent to any complete hyperbolic manifold.

Three dimensional manifolds, Kleinian groups and hyperbolic geometry

- Mathematics
- 1982

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every…