On asymptotic Assouad–Nagata dimension

For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad–Nagata dimension AN-asdim X of X coincides with the covering dimension dim ( ν L X ) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim ( X × R ) = AN-asdim X + 1 . We note that the similar equality for Gromovʹs asymptotic dimension asdim generally fails to hold [A. Dranishnikov, Cohomological approach to asymptotic dimension, Preprint, 2006]. onally we construct an injective map ξ : cone ω ( X ) ∖ [ x 0 ] → ν L X from the asymptotic cone without the basepoint to the sublinear Higson corona.