Local-to-asymptotic topology for cocompact CAT(0) complexes. (Q1398198)

A connectivity at infinity property is one of the asymptotic invariants of discrete groups. The authors generalize connectivity at infinity properties in a finite, non-positively curved cubical complex to an arbitrary finite, non-positively curved complex \(X\). If for each cell \(\sigma\) in \(X\) and for each point \(\rho\) in \(LK(\sigma)\), the spaces \(PLK(\sigma,\rho)\) are (\(n-|\sigma|)\)-acyclic then the universal cover \(\widetilde X\) is \(n\)-acyclic at infinity and \(H_{n+1}\)-semistable at infinity. Moreover, if the links and punctured links of vertices are all 1-connected, then \(\widetilde X\) is \(n\)-connected at infinity and \(\pi_{n+1}\)-semistable at infinity. In particular, the Examples 4.1 and 4.2 are very interesting, and suggest that asymptotic-to-local results are not easy to get.