# On semistability of CAT(0) groups

### Ross Geoghegan

Binghamton University (SUNY), USA### Eric Swenson

Brigham Young University, Provo, USA

## 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 $X,∂X$ 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 $∂X$ 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 $2π $ about that weak cut point is a “cut set” in the sense that it separates $∂X$. Finally, we observe that if a negative example $(G,X)$ exists then $G$ is rank 1.

## Cite this article

Ross Geoghegan, Eric Swenson, On semistability of CAT(0) groups. Groups Geom. Dyn. 13 (2019), no. 2, pp. 695–705

DOI 10.4171/GGD/501