# 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…

## 11 Citations

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## References

SHOWING 1-10 OF 101 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…

### Intersection Form, Laminations and Currents on Free Groups

- Mathematics
- 2007

Let F be a free group of rank N ≥ 2, let μ be a geodesic current on F and let T be an $${\mathbb{R}}$$-tree with a very small isometric action of F. We prove that the geometric intersection number…

### 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…

### Rips Induction: Index of the dual lamination of an $\R$-tree

- Mathematics
- 2010

Let $T$ be a $\R$-tree in the boundary of the Outer Space CV$_N$, with dense orbits. The $Q$-index of $T$ is defined by means of the dual lamination of $T$. It is a generalisation of the…