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Friday, October 16, 2015

**Abstract:** Cannon's conjecture is one of the main open problems in geometric group theory. It states that every Gromov hyperbolic group with boundary topologically equivalent to the 2-sphere acts geometrically on hyperbolic 3-space. It turns out that this conjecture is equivalent to a uniformization problem in analysis on metric spaces - that every Gromov hyperbolic group with boundary topologically equivalent to the 2-sphere has its boundary quasisymmetrically equivalent to the 2-sphere. We will identify the major players in these two conjectures and outline a proof of their equivalence.