On semistability of CAT(0) groups

  • Ross Geoghegan

    Binghamton University (SUNY), USA
  • Eric Swenson

    Brigham Young University, Provo, USA
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Does every one-ended CAT(0) group have semistable fundamental group at infinity? As we write, this is an open question. Let be such a group acting geometrically on the proper CAT(0) space . 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 are properly homotopic. We then show that if the answer to the question is negative, with a counter-example, then the boundary of 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 where 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 about that weak cut point is a “cut set” in the sense that it separates . Finally, we observe that if a negative example exists then 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