On the Higson corona of uniformly contractible spaces (Q1382172)

Let \(X\) be a proper metric space and let \(\nu X\) be its Higson corona. The authors prove that the dimension \(\dim\nu X\) does not exceed the asymptotic dimension of \(X\). In particular, for an hyperbolic space \(X\), \(\dim\nu X<\infty\). This result is interesting with regard to the result of G. Yu: Let \(G\) be a group with a finite complex BG. If the asymptotic dimension of \(G\) is finite, then the Novikov conjecture holds for \(\Gamma\). The authors prove that for finitely generated groups \(\Gamma'\subset\Gamma\) the inequality \(\dim\nu \Gamma'\leq \dim\nu \Gamma\) holds. They also show that the Weinberger conjecture holds for \(EG\) if \(EG\) admits a compactification \(\overline X\) that is rationally acyclic and such that the action of \(G\) on \(EG\) is small at infinity in \(\overline X\).