[Paper Review] Cannon-Thurston Maps for Surface Groups II: Split Geometry and the Minsky Model

Mahan Mj|arXiv (Cornell University)|Jul 20, 2006

Geometric and Algebraic Topology30 references10 citations

TL;DR

This paper proves the existence of Cannon-Thurston maps for all finitely generated Kleinian groups by establishing that all surface groups admit split geometry via Minsky’s model, thereby confirming a conjecture of McMullen and extending earlier results on locally connected limit sets. The work resolves a long-standing question posed by Cannon and Thurston regarding the structure of limit sets in surface group dynamics.

ABSTRACT

In earlier work, we had shown that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map. We had also generalised this result to 3-manifolds whose cores have boundary that is incompressible away from cusps. Here, we show that all surface groups enjoy split geometry by using Minsky’s model for such groups. In combination with our earlier work, this answers a question (conjecture) raised by Cannon and Thurston. We also extend this to show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups. This proves a conjecture of McMullen. It follows that connected limit sets of Kleinian

Motivation & Objective

• To establish that all surface groups exhibit split geometry, resolving a key step in understanding their geometric and dynamical structure.
• To extend the existence of Cannon-Thurston maps from surface groups with split geometry to all finitely generated Kleinian groups.
• To confirm the conjecture of McMullen on the existence of Cannon-Thurston maps for arbitrary finitely generated Kleinian groups.
• To resolve the Cannon-Thurston conjecture regarding the local connectedness of limit sets for surface groups.
• To unify the geometric model of Minsky with dynamical properties of limit sets in Kleinian group theory.

Proposed method

• Utilizes Minsky’s geometric model for surface groups to analyze their quasi-isometric and dynamical properties.
• Applies the theory of split geometry to show that all surface groups satisfy the geometric conditions required for the existence of Cannon-Thurston maps.
• Constructs a natural Cannon-Thurston map as a continuous, equivariant extension from the boundary of the surface group to the limit set of the Kleinian representation.
• Leverages the incompressibility of boundary components (away from cusps) in 3-manifold cores to generalize the result to broader classes of Kleinian groups.
• Employs topological and geometric techniques to prove that the limit set is locally connected under the constructed map.
• Extends the argument from surface groups to arbitrary finitely generated Kleinian groups by analyzing their geometric and dynamical behavior.

Experimental results

Research questions

• RQ1 Do all surface groups admit split geometry, as required for the existence of Cannon-Thurston maps?
• RQ2 Can the existence of Cannon-Thurston maps be extended from surface groups to all finitely generated Kleinian groups?
• RQ3 Does the limit set of any finitely generated Kleinian group remain locally connected under the Cannon-Thurston map?
• RQ4 Is Minsky’s geometric model sufficient to establish the dynamical and topological properties needed for Cannon-Thurston maps in general?
• RQ5 Can the conjecture of Cannon and Thurston on the structure of limit sets be fully resolved using split geometry and model theory?

Key findings

• All surface groups admit split geometry, as established through Minsky’s geometric model for such groups.
• Cannon-Thurston maps exist for all finitely generated Kleinian groups, confirming a conjecture of McMullen.
• The limit set of any surface group is locally connected, as a consequence of the constructed Cannon-Thurston map.
• The result generalizes to 3-manifolds whose cores have incompressible boundary away from cusps, extending earlier findings.
• The existence of a continuous, equivariant extension from the boundary of the surface group to the limit set confirms the topological regularity of the limit set.
• The work provides a complete resolution to the Cannon-Thurston conjecture regarding the local connectedness of limit sets in surface group dynamics.