Embedding amenable groups into a hyperbolic space
20.30 SR 2.058
It is well-known that a graph that quasi-isometrically embeds into a hyperbolic space must be hyperbolic. For instance if an amenable group quasi-isometrically embeds into a hyperbolic space, then it must be virtually cyclic. Coarse embeddings are much more flexible: for instance any abelian, and more generally any nilpotent group coarsely embeds into some hyperbolic group. We prove a sort of converse: any amenable group that coarsely embeds into a hyperbolic group must be virtually nilpotent. The proof uses a variety of different tools, including analysis and ergodic theory.