On the Čech cohomology of Morse boundaries

We consider cusped hyperbolic n–manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most n−3 and does not vanish in dimension n−2. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank–1 simple Lie groups.