  In this note we prove that if a finitely generated amenable group admits a
regular map to a direct product of a hyperbolic space and a euclidean space,
then it must be virtually nilpotent. We deduce that an amenable group regularly
embeds into a hyperbolic group if and only if it is virtually nilpotent,
answering a question of Hume and Sisto. We describe an application to Lorentz
geometry due to Charles Frances.
