Boundaries of cocompact proper CAT(0) spaces

Abstract

A proper CAT(0) metric space $X$ is cocompact if it has a compact generating domain with respect to its full isometry group. Any proper CAT(0) space, cocompact or not, has a compact metrizable boundary at infinity ${\partial }_{\infty }X$; indeed, up to homeomorphism, this boundary is arbitrary. However, cocompactness imposes restrictions on what the boundary can be. Swenson showed that the boundary of a cocompact $X$ has to be finite-dimensional. Here we show more: the dimension of ${\partial }_{\infty }X$ has to be equal to the global Čech cohomological dimension of ${\partial }_{\infty }X$. For example: a compact manifold with non-empty boundary cannot be ${\partial }_{\infty }X$ with $X$ cocompact. We include two consequences of this topological/geometric fact: (1) The dimension of the boundary is a quasi-isometry invariant of CAT(0) groups. (2) Geodesic segments in a cocompact $X$ can “almost” be extended to geodesic rays, i.e. $X$ is almost geodesically complete.