Abstract:A Hyperbolic space is a metric space whose geodesic triangles are "thin" and a hyperbolic group is a group whose Cayley graph is hyperbolic. We focus on the boundary of hyperbolic groups in this talk and go through topics around two rigidity conjectures: The Cannon Conjecture and the Kapovich-Kleiner Conjecture. Roughly speaking, these two conjectures ask about whether special topological restrictions on the boundary of hyperbolic groups will uniquely determine them up to quasisymmetries. In an effort to answer these questions, many people have studied them from different approaches. We will go through the work by Bonk and Kleiner. In the end, we provide our trying on these conjectures. In particular, we have constructed a special case of metric Sierpinski carpet, dyadic slit carpets, and completely characterize its planar quasisymmetric embeddability.