Volumes of cusped hyperbolic manifolds

For n-dimensional hyperbolic manifolds of finite volume with m ⩾ 1 cusps a new lower volume bound is presented which is sharp for n = 2,3. The estimate depends upon m and the ideal regular simplex volume. The proof makes essential use of a density argument for ball packings in Euclidean and hyperbolic spaces and explicit formulae for the simplicial density function. Examples, consequences for the Gromov invariant, and-for n even-the maximal number of cusps are discussed.