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 H^n× R^d , 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 Zⁿ⁺ᵈ⁻¹ coarsely embeds into H n × R d H^n× R^d . 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].