6 J ul 2 00 6 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 Gromov's asymptotic dimension asdim generally fails to hold [Dr3]. Additionally we construct an injective map ξ : cone ω (X) \ [x 0 ] → ν L X from the asymp-totic cone without the base point to the sublinear Higson corona. §1 Introduction The Assouad-Nagata dimension was introduced in the 80s by Assouad [As1],[As2] under the name Nagata dimension. Recently this notion was revived in the asymptotic geometry due to works of Lang and Schlichenmaier [LSch], and Buyalo and Lebedeva [Bu], [BL]. The concept takes into account the dimension of a metric space on all scales. In this paper we consider only the large scale version of it. Note that the asymptotic version of the Assouad-Nagata dimension agrees with the original for our main source of examples of metric spaces-finitely generated discrete groups with the word metric. Like in the case of Gromov's asymptotic dimension, the Assouad-Nagata dimension is a group invariant. A certain analogy between the asymptotic Assouad-Nagata dimension AN-asdim and the asymptotic dimension asdim invites one to transfer the asymptotic dimension theory