On coarse embeddings of amenable groups into hyperbolic graphs

In this note we prove that if a finitely generated amenable group admits a regular map to H n × R d Hⁿ Rᵈ, then it must be virtually nilpotent of degree of growth at most d + n − 1 d+n-1. This is sharp as Z n + d − 1 Z^n+d-1 coarsely embeds into H n × R d Hⁿ Rᵈ. We deduce that an amenable group regularly (or coarsely) embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto New York J. Math. 23 (2017), pp. 1657–1670. We describe an application to Lorentz geometry due to Charles Frances Geom. Funct. Anal. 31 (2021), pp. 1095–1159.