# Conical limit points and the Cannon-Thurston map

@article{Jeon2014ConicalLP, title={Conical limit points and the Cannon-Thurston map}, author={Woojin Jeon and Ilya Kapovich and Christopher J. Leininger and Ken'ichi Ohshika}, journal={arXiv: Group Theory}, year={2014} }

Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in $Z$ in terms of their pre-images under the Cannon-Thurston map $i$. As an application we prove, under the extra assumption that the action of $G$ on $Z$ has no accidental…

## 10 Citations

Measurable rigidity for Kleinian groups

- MathematicsErgodic Theory and Dynamical Systems
- 2015

Let $G,H$ be two Kleinian groups with homeomorphic quotients $\mathbb{H}^{3}/G$ and $\mathbb{H}^{3}/H$ . We assume that $G$ is of divergence type, and consider the Patterson–Sullivan measures of $G$…

Cannon–Thurston maps for hyperbolic free group extensions

- Mathematics
- 2015

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a…

Corrigendum to “Graphs of hyperbolic groups and a limit set intersection theorem”

- MathematicsProceedings of the American Mathematical Society
- 2021

We say that a collection $\mathcal S$ of subgroups of a hyperbolic group $G$ satisfies the limit set intersection property if for any $H, K\in \mathcal S$ we have $\Lambda(H)\cap…

CANNON–THURSTON MAPS

- Mathematics
- 2018

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch…

A universal Cannon-Thurston map and the surviving curve complex

- MathematicsTransactions of the American Mathematical Society, Series B
- 2022

Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve…

Horospheres in degenerate 3-manifolds

- Mathematics
- 2016

We study horospheres in hyperbolic 3-manifolds M all whose ends are degenerate. Deciding which horospheres in M are properly embedded and which are dense reduces to a) studying the horospherical…

Cannon–Thurston maps in Kleinian groups and geometric group theory

- MathematicsSurveys in Differential Geometry
- 2020

We give a survey account of Cannon-Thurston maps, both in the original context of Kleinian groups, as well as in the more general context of Geometric Group Theory. Some of the principal applications…

Cannon–Thurston fibers for iwip automorphisms of FN

- MathematicsJ. Lond. Math. Soc.
- 2015

It is proved that for any $\phi$, the map $\hat \iota$ is finite-to-one and that the preimage of every point of $\partial G_\phi$ has cardinality $\le 2N$.

Geometric group theory and hyperbolic geometry: Recent contributions from Indian mathematicians

- MathematicsIndian Journal of Pure and Applied Mathematics
- 2019

Geometric group theory emerged as a distinct branch of mathematics through the seminal work of Gromov [25] in 1987 and since then it has been a very active area of research intermingling with many…

CANNON–THURSTON MAPS

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch…

## References

SHOWING 1-10 OF 119 REFERENCES

Floyd maps for relatively hyperbolic groups

- Mathematics
- 2012

Let $${\mathbf{delta}_{\mathcal S,\lambda}}$$ denote the Floyd metric on a discrete group G generated by a finite set $${\mathcal S}$$ with respect to the scaling function fn = λn for a positive λ <…

Measurable rigidity for Kleinian groups

- MathematicsErgodic Theory and Dynamical Systems
- 2015

Let $G,H$ be two Kleinian groups with homeomorphic quotients $\mathbb{H}^{3}/G$ and $\mathbb{H}^{3}/H$ . We assume that $G$ is of divergence type, and consider the Patterson–Sullivan measures of $G$…

Axes in Outer Space

- Mathematics
- 2006

The authors develop a notion of axis in the Culler-Vogtmann outer space $\mathcal{X}_r$ of a finite rank free group $F_r$, with respect to the action of a nongeometric, fully irreducible outer…

The Cannon–Thurston map for punctured-surface groups

- Mathematics
- 2006

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that…

Cannon–Thurston maps for hyperbolic free group extensions

- Mathematics
- 2015

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a…

IRREDUCIBLE AUTOMORPHISMS OF $F_{n}$ HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2003

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are…

Geometric Intersection Number and analogues of the Curve Complex for free groups

- Mathematics
- 2009

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ : \bar{cv}(F_N)\times Curr(F_N)\to…

INVARIANT LAMINATIONS FOR IRREDUCIBLE AUTOMORPHISMS OF FREE GROUPS

- Mathematics
- 2011

For every atoroidal iwip automorphism $\phi$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(\phi)$ is…

Similar Relatively Hyperbolic Actions of a Group

- Mathematics
- 2013

Let a discrete group G possess two convergence actions by homeomorphisms on compacta X and Y . Consider the following question: does there exist a convergence action GyZ on a compactum Z and…