[Paper Review] Conical limit points and the Cannon-Thurston map

Woojin Jeon, Ilya Kapovich+2 more|arXiv (Cornell University)|Jan 12, 2014

Geometric and Algebraic Topology69 references3 citations

TL;DR

This paper provides dynamical and geometric characterizations of conical limit points in the boundary of a convergence group action via their pre-images under the Cannon-Thurston map. It proves that if the Cannon-Thurston map is non-injective, then there exists a non-conical limit point with a singleton pre-image, extending known results in hyperbolic group theory and Kleinian group dynamics.

ABSTRACT

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 $i:\partial G o Z$, 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 parabolics, that if the map $i$ is not injective then there exists a non-conical limit point $z\in Z$ with $|i^{-1}(z)|=1$. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if $G$ is a non-elementary torsion-free word-hyperbolic group then there exists $x\in \partial G$ such that $x$ is not a "controlled concentration point" for the action of $G$ on $\partial G$.

Motivation & Objective

• To characterize conical limit points in the boundary of a convergence group action using the pre-images of the Cannon-Thurston map.
• To establish a geometric and a dynamical characterization of conical limit points in terms of their pre-images under the Cannon-Thurston map.
• To prove that if the Cannon-Thurston map is non-injective, then there exists a non-conical limit point with a singleton pre-image, under the no-accidental-parabolics assumption.
• To apply these results to natural contexts such as subgroups of word-hyperbolic groups and Kleinian surface groups.
• To show the existence of non-controlled concentration points in the boundary of torsion-free, non-elementary word-hyperbolic groups.

Proposed method

• Utilizes the existence of a continuous, equivariant Cannon-Thurston map $ i: ar{ ho} o Z $ from the boundary of a word-hyperbolic group $ G $ to a compact metrizable space $ Z $, where $ G $ acts as a convergence group.
• Applies the theory of conical limit points in convergence group actions, defined via sequences converging to the boundary with uniform quasi-geodesic tracking.
• Analyzes the pre-image structure of the Cannon-Thurston map, particularly focusing on points in $ Z $ with singleton pre-images.
• Employs results from Mitra on ending laminations and the dynamics of automorphisms on free group boundaries to understand limit point behavior.
• Applies the theory of hyperbolic extensions of free groups and the structure of stable laminations $ L_{BFH}( heta) $ to analyze the image of the Cannon-Thurston map.
• Uses the fact that for $ G = F_N times_ heta bZ $, the image of the Cannon-Thurston map is $ L(T_-) igcup L(T_+) $, the union of dual laminations to the attracting and repelling trees.

Experimental results

Research questions

• RQ1 What characterizations of conical limit points in $ Z $ can be given in terms of their pre-images under the Cannon-Thurston map?
• RQ2 Under what conditions does the non-injectivity of the Cannon-Thurston map imply the existence of a non-conical limit point with a singleton pre-image?
• RQ3 How do the dynamics of the action of $ G $ on $ Z $, particularly in the absence of accidental parabolics, affect the structure of limit points?
• RQ4 What is the relationship between the ending lamination data of an automorphism and the image of the Cannon-Thurston map in hyperbolic extensions?
• RQ5 Can one guarantee the existence of non-controlled concentration points in the boundary of a torsion-free, non-elementary word-hyperbolic group?

Key findings

• The paper establishes a dynamical characterization of conical limit points in $ Z $: a point $ z o Z $ is conical if and only if its pre-image under $ i $ contains a pair of distinct points that are strongly transverse in the boundary of $ G $.
• It provides a geometric characterization: $ z o Z $ is conical if and only if the pre-image $ i^{-1}(z) $ is a pair of points that are not linked by any $ G $-invariant lamination.
• If the Cannon-Thurston map $ i $ is not injective and the action of $ G $ on $ Z $ has no accidental parabolics, then there exists a non-conical limit point $ z o Z $ such that $ |i^{-1}(z)| = 1 $.
• For hyperbolic extensions $ G = F_N times_ heta bZ $ with $ heta $ fully irreducible and atoroidal, the image of the Cannon-Thurston map is $ L(T_-) igcup L(T_+) $, the union of the dual laminations to the attracting and repelling trees.
• In the case of torsion-free, non-elementary word-hyperbolic groups, there exists a boundary point $ x o ar{ ho} $ that is not a controlled concentration point for the action on $ ar{ ho} $.
• The paper confirms that the image of the Cannon-Thurston map in the case of surface group actions on $ bH^3 $ is the union of the stable and unstable laminations, consistent with known results in Kleinian group theory.