We give new examples of hyperbolic and relatively hyperbolic
groups of cohomological dimension $d$ for all $d ≥ 4$ (see Theorem
2.13). These examples result from applying $CAT(0)/CAT(−1)$
filling constructions (based on singular doubly warped products)
to finite volume hyperbolic manifolds with toral cusps.
¶ The groups obtained have a number of interesting properties,
which are established by analyzing their boundaries at infinity by
a kind of Morse-theoretic technique, related to but distinct from
ordinary and combinatorial Morse theory (see Section 5).