6.31 (1978)

a) Suppose that $G$ is a finitely-generated residually-finite group, $d(G)$ the minimal number of generators of $G$, and $\delta(G)$ the minimal number of topological generators of the profinite completion of $G$. Is $\delta(G) = d(G)$ always true?
b) Let $G$ be a finitely generated residually finite group, $d(G)$ the minimal number of its generators, and $\delta(G)$ the minimal number of topological generators of the profinite completion of $G$. It is known that there exist groups $G$ for which $d(G) > \delta(G)$ (G. A. Noskov, Math. Notes, 33, no. 4 (1983), 249–254). Is the function $d$ bounded on the set of groups $G$ with the fixed value of $\delta(G) \geqslant 2$?