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Let $G$ be a profinite group of rank $d \geq 2015$, which is a fibered product of two groups of rank at most $2$. That is, there exist closed normal subgroups $N_1, N_2 \lhd_c G$ with $N_1 \cap N_2 = \{1\}$ and $$\max\{d(G/N_1), d(G/N_2)\} \leq 2.$$

Schreier's bound for the rank of an open subgroup $U \leq_o G$ is: $$d(U) \leq [G:U](d-1)+1.$$

Can we beat Schreier's bound in this case? That is:

Must there be some open subgroup $K \leq_o G$ with $d(K) < [G:K](d-1)+1$ ?

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  • $\begingroup$ Why $d \geq 2015$? $\endgroup$
    – Stefan Kohl ♦
    Commented May 24, 2015 at 15:37
  • $\begingroup$ @StefanKohl I am ready to assume that $d$ is arbitrarily big, but I am not sure how big should it be. Maybe even $d \geq 6$ is good enough. $\endgroup$
    – Pablo
    Commented May 24, 2015 at 15:49

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