On semistability of $CAT(0)$ groups

TL;DR

This paper investigates the open question of whether all one-ended $CAT(0)$ groups have semistable fundamental groups at infinity, linking this to properties of geodesic rays and boundary topology.

Contribution

It reduces the problem to checking proper homotopy of geodesic rays and explores boundary conditions, highlighting the role of weak cut points in potential counterexamples.

Findings

Proper homotopy of geodesic rays is key to semistability.

Counterexamples, if any, must have boundaries with weak cut points.

Any negative example would involve a rank 1 group.

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 $X$, $\del X$ with the cone topology, must have a weak cut point. This is of interest because a theorem of Papasoglu and the second-named author \cite{PS} has established that there cannot be an example of $(G,X)$ where $\del X$ 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 $\del X$. Finally, we observe that if a negative example $(G, X)$ exists then $G$ is rank 1.