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Let $M$ be a closed, minimal, hypersurface in the sphere $S^{n-1}$, $n\geq 4$. Suppose $M$ has $H^1(M,\mathbb{Z})=0$. What can we say about the cardinality of the first fundamental group of $M$, $\pi_1(M)$?

By duality and Hurewicz theorem, we know that $\pi_1(M)$ is perfect. So, in terms of group theory, we are asking whether there exist a finitely presented perfect group with infinitely many elements. But of course, we have the aditional information that M is compact minimal in $S^{n-1}$.

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    $\begingroup$ An example of an infinite f.p. perfect group is $SL_3(\mathbf{Z})$. Or quite equivalently the group with presentation $\langle e_{ij}:1\le i\neq j\le 3\mid [e_{ij},e_{jk}]=e_{ik}: \forall i\neq j\neq k\neq i\rangle$. $\endgroup$
    – YCor
    Commented Jul 29, 2019 at 15:35
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    $\begingroup$ @YCor OK, you're right. I deleted my suggestion. It's still not clear if the OP wants an infinite fp perfect group in the abstract, or one that arises as above. $\endgroup$
    – Donu Arapura
    Commented Jul 29, 2019 at 15:42
  • $\begingroup$ OK. I'll suggest the OP to reformulate the question so as to focus on the question with the additional requirement. (Btw, there are perfect cocompact Fuchsian groups, and plenty of infinite perfect groups among $\pi_1$ of $3$-dimensional manifolds.) $\endgroup$
    – YCor
    Commented Jul 29, 2019 at 15:50

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