Let \(G\) be a negatively curved group in the sense of M. Gromov [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] and \(P(G)=P_ d(G)\) be the Rips complex for \(G\) with \(d\) as sufficiently large as \(P(G)\) is contractible. Here a connection \(x_ 1,\ddots ,x_ k\in G\) spans a simplex in \(P_ d(G)\) if \(d(x_ i,x_ j)\leq d\) for all \(i,j\) where \(d(x,y)\) is the word metric on \(G\). Considering the boundary \(\partial G\) as the set of equivalence classes of sequences convergent at infinity, \(P(G)\) relates \(\partial G\) with the cohomological properties of \(G\).
The main observation of the paper is: Theorem 1.2: \(\widetilde{P(G)}=P(G)\cup \partial G\) is an absolute retract, and \(\partial G\subset \widetilde{P(G)}\) is a \(\mathbb{Z}\)-set, i.e. for every open set \(U\subset \widetilde{P(G)}\) the inclusion \(U\partial G\hookrightarrow U\) is a homotopy equivalence.
As applications of this result, the authors discuss the local connectivity of \(\partial G\) and prove that the universal covers of closed, irreducible 3-manifolds with infinite negatively curved fundamental group G compactified by \(\partial G\) are homeomorphic to the 3-ball. This is a sharpening of the result of V. Poenaru [J. Differ. Geom. 35, 103-130 (1992)] and A. Casson (unpublished) about homeomorphisms of the universal covers of such 3-manifolds to \(\mathbb{R}^ 3\).