Criterion for Cannon's Conjecture

TL;DR

This paper establishes a criterion linking the structure of hyperbolic groups with boundary homeomorphic to a 2-sphere to Kleinian groups, reducing Cannon's conjecture to the existence of sufficient quasi-convex surface subgroups.

Contribution

It provides a new necessary and sufficient condition for Cannon's conjecture, connecting boundary separation properties to the presence of quasi-convex surface subgroups.

Findings

A hyperbolic group with boundary homeomorphic to a 2-sphere is a Kleinian group if and only if boundary points are separated by quasi-convex surface subgroups.

The criterion reduces Cannon's conjecture to demonstrating the existence of enough such subgroups.

The result offers a new approach to proving Cannon's conjecture through subgroup analysis.

Abstract

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of $G$ are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contains "enough" quasi-convex surface subgroups.