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Let $G$ be a residually finite group, and suppose that for every finite index subgroup $U$ of $G$ we have:

$$d(U) = 2[G : U] + 2.$$

Is $G$ necessarily a surface group?

Here $d(U)$ is the least cardinality of a generating set of the group $U$.

It is also interesting to ask the same question with $d(U)$ replaced by $\dim_{\mathbb{Q}} (H_1(U,\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Q})$.

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    $\begingroup$ Your formula is for $G$ of genus 2? it does not work for other surface groups (the right-hand term does not depend on $d(G$... just taking $U=G$ yields $d(G)=4$.) $\endgroup$
    – YCor
    Mar 21, 2017 at 15:58
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    $\begingroup$ So the question is whether $G$ is isomorphic to $\pi_1$ of a closed surface of Euler characteristic $-2$ (the orientable one, of genus 2, or the non-orientable one, quotient of a genus 3 surface by a fixed-point-free orientation-reversing involution). $\endgroup$
    – YCor
    Mar 21, 2017 at 16:46
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    $\begingroup$ Is there any other group? The answer is already in my previous comment: yes. $\endgroup$
    – YCor
    Mar 21, 2017 at 16:49
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    $\begingroup$ Non-orientable surface groups have a presentation $\langle x,y,z,w|x^2y^2z^2w^2=1\rangle$. You can replace a pair of squares by a commutator, unless you lose the last square this way, the isomorphism type of the group does not change. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Mar 21, 2017 at 19:54
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    $\begingroup$ In the pro-$p$ world there are examples different from surface groups. If $G$ is a 4-generated Demuskin group, then every closed subgroup $U$ needs $2(G:U)+2$ generators. Demuskin groups generalize the pro-$p$ completion of surface groups, as they have presentations of the form $\langle x,y,z,w|x^{p^n}[x,y][z,w]=1\rangle$. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Mar 21, 2017 at 20:04

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