Rank gradient, cost of groups and the rank versus Heegaard genus problem

We study the growth of the rank of subgroups of finite index in residually
finite groups, by relating it to the notion of cost.
As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on
hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in
topological dynamics.