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

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