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Let $\widehat F(k)$ be the free profinite group on $k$ generators and let $p$ be a prime. Then there is a canonical projection $\pi\colon \widehat F(k)\to \widehat F_p(k)$ where $\widehat F_p(k)$ is the free pro-$p$ group on $k$-generators. It is well known that a free pro-$p$ group is projective in the category of profinite groups and so $\pi$ splits via a continuous homomorphism $\psi\colon \widehat F_p(k)\to \widehat F(k)$.

I would like to know if anybody knows an explicit splitting. To be more precise, let $X_k =\{x_1,\ldots, x_k\}$ be a free generating set for $\widehat F(k)$. I would like explicit sequences $\{w_{i,n}\}$, for $i=1,\ldots, k$, of words over $X_k$ such that $w_{i,n}\to w_i$ with $\pi(w_i) = \pi(x_i)$ and $\overline {\langle w_1,\ldots, w_k\rangle}$ a free pro-$p$ group.

For example, if $k=1$, then I know how to write down a sequence converging to an element giving a splitting of $\pi\colon \widehat {\mathbb Z}\to \mathbb Z_p$. Namely, if $x_1$ is the free generator of $\widehat{\mathbb Z}$ and if $p_1,p_2,\ldots$ is a list of the primes other than $p$. Then $x_1^{(p_1p_2\cdots p_n)^{n!}}$ converges to an element generating a copy of $\mathbb Z_p$ and mapping to $\pi(x_1)$.

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  • $\begingroup$ What's a reference for this well-known fact on free pro-$p$-groups among profinite groups? Does it reflect some result about finite groups? $\endgroup$
    – YCor
    Mar 1, 2019 at 19:49
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    $\begingroup$ @YCor, it can be found in Ribes and Zalesskii in the chapter on projective profinite groups. It it is true more generally for extension-closed pseudovarieties of groups and boils down to an argument involving Frattini subgroups and finite groups, and maybe a small amount of cohomology. I forget the details. $\endgroup$ Mar 1, 2019 at 19:55
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    $\begingroup$ @YCor : Here is an argument why free pro-$p$ groups are projective. Let $\pi\colon G \to \hat{F}_p(k)$ be a surjective map from a profinite group onto a free pro-$p$ group. If $W \leq G$ is a Sylow pro-$p$-subgroup, then also $\pi_{|W}$ is surjective (this follows from the same statement on finite groups). Since $\hat{F}_p(k)$ is free pro-$p$, there is a splitting $\psi\colon \hat{F}_p(k) \to W$. $\endgroup$ Mar 14, 2019 at 11:14
  • $\begingroup$ This is basically the argument in Ribes and Zalesskii $\endgroup$ Mar 14, 2019 at 13:51

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