Abstract:
Given a relatively hyperbolic group, the topology of its Bowditch boundary is useful in understanding the underlying relatively hyperbolic structure. For example, for a connected boundary, we know from the work of Hruska–Dasgupta that the cut points are parabolic and correspond to a parabolic splitting of the group. We also know from work of Haulmark–Hruska that inseparable loxodromic cut pairs correspond to splittings over 2–ended subgroups.

For some time, it was suspected that inseparable cut pairs on the boundary can only be loxodromic. However, examples of parabolic cut pairs were discovered by Hruska–Walsh. In this talk, we will present a combination theorem for relatively hyperbolic groups, which results in such parabolic cut pairs. We will then present a decomposition theorem, which states that all relatively hyperbolic groups with inseparable parabolic cut pairs on their boundaries arise from the above-mentioned combination theorem.