Embedding Higson compactification into a product of adelic solenoids (Q6053448)

The notion of Higson compactification was introduced by Higson and Roe for a coarse geometric approach to the Novikov conjecture (see [\textit{J. Roe}, Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043); Lectures on coarse geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)]). In this paper, the authors prove that every simply connected proper geodesic metric space \(X\) admits an embedding of its Higson compactification into a product of adelic solenoids that induces an isomorphism of 1-dimensional integral Čech cohomology; and that for each group \(\Gamma\) with a finite classifying CW complex \(B\Gamma\), the universal covering \(E\Gamma\) taken with the lifted metric admits a \(\Gamma\)-equivariant embedding of its Higson compactification into a product of adelic solenoids that induces an epimorphism of the 1-dimensional integral Čech cohomology. These theorems are related to a result by \textit{A. N. Dranishnikov} et al. [Commun. Pure Appl. Math. 61, No. 2, 139--155 (2008; Zbl 1137.57027)]. The authors also give embedding theorems of Higson compactifications into a product of \(p\)-adic solenoids and one of Knaster continua.