A converse to a splitting theorem

In a joint paper with M. Mihalik, a theorem of the following type was proved: If G is a one-ended group acting geometrically on a CAT(0) space X and it is known that G splits as an amalgamated product in a “special way” (to be
described in the talk), then the visual boundary of X is connected but not locally connected. In joint work in progress with C. Hruska, we consider a full converse to this theorem in the setting of groups acting on CAT(0) spaces with
the Isolated Flats Property. In this talk, we will discuss several motivating examples in detail. The groups in question here are known to be relatively hyperbolic with respect to the stabilizers of flats, thus one can also consider the Bowditch boundary of such a group. We discuss the relationship between the visual boundary of the space X and the Bowditch boundary of the group G; this is crucial in our approach to the converse of the splitting theorem.