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.
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Miklós Abért, Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1657–1677DOI 10.4171/JEMS/344