  We introduce an obstruction to the existence of a coarse embedding of a given
group or space into a hyperbolic group, or more generally into a hyperbolic
graph of bounded degree. The condition we consider is "admitting exponentially
many fat bigons", and it is preserved by a coarse embedding between graphs with
bounded degree. Groups with exponential growth and linear divergence (such as
direct products of two groups one of which has exponential growth, solvable
groups that are not virtually nilpotent, and uniform higher-rank lattices) have
this property and hyperbolic graphs do not, so the former cannot be coarsely
embedded into the latter. Other examples include certain lacunary hyperbolic
and certain small cancellation groups.
