On the Higson corona of uniformly contractible spaces

Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdimX of X introduced by M. Gromov. In particular, it implies that dim νRn = n for euclidean and hyperbolic metrics on Rn. We prove that for finitely generated groups Γ′ ⊃ Γ with word metrics the inequality dim νΓ′ ⩽ dim νΓ holds. Also we prove that a small action at infinity of a geometrically finite group Γ on some compactification X′ of the universal covering space X = EΓ enables one to map the Higson compactification onto X′. In that case the rational acyclicity of X′ implies the conjecture by S. Weinberger for X which is a form of the Novikov Conjecture for Γ.