  We show that an infinite residually finite boundedly generated group has an
infinite chain of finite index subgroups with ranks uniformly bounded, and give
(sublinear) upper bounds on the ranks of arbitrary finite index subgroups of
boundedly generated groups (examples which come close to achieving these bounds
are presented). This proves a strong form of a conjecture of Abert,
Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite
boundedly generated residually finite groups is $0$. Furthermore, our first
result establishes a variant of a conjecture of Lubotzky on the ranks of finite
index subgroups of special linear groups over the integers, and is analogous to
a result of Pyber and Segal for solvable groups.
