2004/07/29 by Ross Geoghegan, Geoghegan, Ross, Pedro Ontaneda +1
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Metric Geometry (math.MG) #math.AT #math.MG
paper · pdf · doi:10.48550/arxiv.math/0407506
arxiv created 2004/07/29 · arxiv updated 2009/12/01
This paper is about geometric and topological properties of a proper CAT(0) space X 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 quasi- isometry 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.)