Criterion for Cannon's Conjecture

Vladimir Markovic

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arXiv · 1205.5747

Criterion for Cannon's Conjecture

Vladimir Markovic

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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.

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Vladimir Markovic. 2012-05-25. Criterion for Cannon's Conjecture. https://arxiv.org/abs/1205.5747

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