Cocompact Proper CAT(0) Spaces

This paper is about geometric and topological properties of a proper CAT(0) spaceX which is cocompact i.e. which has a compact generating domain with respect to the full isometry group. It is shown that geodesic segments in X can “almost” be extended to geodesic rays. A basic ingredient of the proof of this geometric statement is the topological theorem that there is a top dimension d in which the compactly supported integral cohomology of X is non-zero. It is also proved that the boundary-at-infinity of X (with the cone topology) has Lebesgue covering dimension d− 1. It is not assumed that there is any cocompact discrete subgroup of the isometry group of X ; however, a corollary for that case is that “the dimension of the boundary” is a quasiisometry invariant of CAT(0) groups. (By contrast, it is known that the topological type of the boundary is not unique for a CAT(0) group.) 1 Statement of Theorems A CAT(0) space is a geodesic metric space (X, dX ) whose geodesic triangles are “no fatter than” the corresponding comparison triangles in the Euclidean plane. A general reference for facts about CAT(0) spaces used here is [4]. We will usually suppress dX referring just to X. Such a space X is proper if all closed balls are compact, and is cocompact if there is a compact generating domain for the full isometry group of X, i.e. there is a compact set C ⊂ X such that the sets {h(C) | h is an isometry of X} cover X. In particular, a proper CAT(0) space X has a compact boundary, ∂∞X, namely the set of asymptoty classes ∗The second-named author was supported in part by a research grant from CAPES, Brazil.