Computing higher limits over the fusion orbit category via amalgams

We study higher limits over the centric orbit category of a fusion system realized by an amalgamated product. In so doing we provide a novel technique for studying the Diaz-Park sharpness conjecture and prove it (in the case of the cohomology Mackey functors) for all the Clelland-Parker and Parker-Stroth fusion systems. This complements previous work from Henke, Libmand and Lynd. We further use the developed technique to study the Benson-Solomon fusion systems thus relating higher limits over the centric fusion orbit category of these systems with the signalizer functors described by Aschbacher and Chermak. We believe that the proposed technique can, in future work, be used as a first step in an induction argument that can bring us closer to providing an answer to this conjecture.