The boundary of negatively curved groups

Gromov's article Gr contains fundamental properties of negatively curved groups. Several sets of seminar notes are available FrN, SwN, USN that contain more detailed accounts of, and further expand on, Gromov's article. A sequence Xi E r, i = 1, 2,. . . , is convergent at infinity provided (Xi' x) -t 00 as i, j -t 00. Two sequences Xi' YJ convergent at infinityare equivalent if (Xi' Yi) -t 00 as i -t 00. A point in the boundary ar of r is an equivalence class of sequences convergent at infinity. Now r u ar has a natural topology, in which r is a discrete subspace, and a typical (not necessarily open) neighborhood of a point a E ar represented by a sequence {xJ is given by {y E r u arj (a. y) > N, where (a· y) = lim (xi. y) if y E r, and (a· y) = lim (xi. Yi) if YEar is represented by a sequence {yJ. ar is a compact, metrizable, finite-dimensional space Gr, SwN. The important tool that relates ar with the cohomological properties of r is the Rips complex Pd (r). For every d ~ 0, Pd (r) is the simplicial complex whose vertices are elements of r, and a collection XI' •••, xk E r spans a simplex if d (Xi' x) ~ d for all i, j. The natural group action of r on itself by left translations gives rise to an action on Pd (r). The key observation of Rips is that when d is sufficiently large, Pd (r) is contractible and therefore, when r is torsion free, provides a model for Er. More precisely, we have Proposition 1. 1 Gr, Lemma 1. 7. A; SwN, §4. 2, Proposition 9. Suppose that r is a negatively curved group, and let t5 be as in the definition. Choose an integer