  We prove that the cardinality of the torsion subgroups in homology of a
closed hyperbolic manifold of any dimension can be bounded by a doubly
exponential function of its diameter. It would follow from a conjecture by
Bergeron and Venkatesh that the order of growth in our bound is sharp.
  We also determine how the number of non-commensurable closed hyperbolic
manifolds of dimension at least 3 and bounded diameter grows. The lower bound
implies that the fraction of arithmetic manifolds tends to zero as the diameter
goes up.
