CAT(0) HNN-extensions with non-locally connected boundary

The main result of this paper shows that when a certain type of HNN-extension acts geometrically on a CAT(0) space, the boundary of the space is not locally connected. The motivating example is an HNN-extension that arises from studying parabolic semidirect products of F2 and Z. We show all such semidirect products act on the CAT(0) space constructed here. We explicitly identify a point of non-local connectivity in the boundary of this space as a guide to reading the proof of the main theorem.