On semistability of CAT(0) groups

Abstract

Does every one-ended CAT(0) group have semistable fundamental group at infinity? As we write, this is an open question. Let $G$ be such a group acting geometrically on the proper CAT(0) space $X$. In this paper we show that in order to establish a positive answer to the question it is only necessary to check that any two geodesic rays in $X$ are properly homotopic. We then show that if the answer to the question is negative, with $(G,X)$ a counter-example, then the boundary of $X,\partial X$ with the cone topology, must have a weak cut point. This is of interest because a theorem of Papasoglu and the second-named author [10] has established that there cannot be an example of $(G,X)$ where $\partial X$ has a cut point. Thus, the search for a negative answer comes down to the difference between cut points and weak cut points. We also show that the Tits ball of radius $\frac{\pi }{2}$ about that weak cut point is a “cut set” in the sense that it separates $\partial X$. Finally, we observe that if a negative example $(G,X)$ exists then $G$ is rank 1.