Counting hyperbolic manifolds with bounded diameter

Let ρ _n (V) be the number of complete hyperbolic manifolds of dimension n with volume less than V. Burger et al [Geom. Funct. Anal. 12(6) (2002), 1161-1173.] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV log V ≤ log ρ _n (V) ≤ bV log V, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially with volume. Additionally, this bound holds in dimension 3.