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Abstract

An important theme in geometric group theory is to study groups up to coarse equivalences. Quasiisometry is one of the most important examples of such an equivalence.

Boundaries of word hyperbolic groups equipped with their quasi-conformal strutures are full quasi-isometric invariants of groups. This provides a very strong link between geometry and (nonsmooth) analysis, which has been very active recently.

We will describe these objects and discuss the Cannon's conjecture which is one of the main problems inhe subject. We will present a strategy due to Bonk and Kleiner to approach this conjecture.