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Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is anything analogous for its profinite completion.

Namely, let $\hat{F}_r$ be the free profinite group on $r$ generators. Then $\hat{F}_r$ contains the free group $F_r$ as a dense subgroup, and every element of $\hat{F}_r$ can be written as an infinite product $\prod_{i=1}^\infty x_i$ where each $x_i \in F_r$ and the $x_i$ tend to $1$ in the profinite topology.

In other words, we can arrange that $x_i \to 1$ in the following sense: for every $N$, the $x_i$ have the property that almost all of them are mapped to $1$ by every homomorphism $F_r \to G$ where $G$ is any group of cardinality $\leq N$ (for $i \gg 0$, depending on $N$). This suggests a natural descending filtration on $F_r$ via the intersection of the kernels of all maps $F_r \to G$ where $|G|\leq N$. Given a nice description (e.g., generators) of the elements in this filtration, one would have a somewhat explicit procedure of describing elements in the profinite completion via "infinite" words.

There are some natural elements in filtration $N$ in $F_r$, namely the $c(N)$th powers where $c(N)$ is the least common multiple of $\{1, 2, \dots, N\}$.

Question: Is the $N$th filtration in $F_r$ generated by the $c(N)$th powers? That is, if $a \in F_r$ is an element that goes to $1$ in every finite group of cardinality $\leq N$ a (finite) product of $c(N)$th powers?

Question${}^\prime$: If the above is too much to expect, are the two possible filtrations on $F_r$ above at least "commensurable"?

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If n is a number sufficiently large, then the (normal) subgroup generated by n-th powers is not finite index by the Burnside problem. –  Benjamin Steinberg Sep 29 '13 at 1:30

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up vote 7 down vote accepted

Let me answer the question in the title rather than in the body. First of all for the free profinite group on one generator $\widehat Z$ we have a direct product of p-adic integers over all primes and so we know how to write down elements. Now if $w$ is any element of a profinite group and $\lambda\in \widehat Z$ then define $w^\lambda$ to be the image of $\lambda$ under the map sending the generator of $\widehat Z$ to $w$. So we can build up some elements by starting with words and closing under products and infinite powers.

Here is an example. Let $p$ be a prime. The sequence $p^{n!}$ converges in $\widehat Z$ to a generator, denoted $p^{\omega}$, of the $p'$-component (i.e., the product of all q-adic groups with $q\neq p$). Hence a profinite group is pro-$p$ if and only if it satisfies the profinite identity $x^{p^{\omega}}=1$ because $x^{p^{\omega}}$ generates the p'-prime component of the pro-cyclic subgroup generated by $x$.

Jorge Almeida, principally in the context of free profinite monoids, but also for free profinite groups, came up with the following method to generate further elements. Let $M=End(\widehat F_r)$ be the endomorphism monoid. It is a profinite monoid. Given any endomorphism $\phi$ one has that $\phi^{\omega}=\lim_{n\to \infty} \phi^{n!}$ is an idempotent endomorphism. One can create interesting and useful elements which are recursive (in several senses, eg,there is an re sequence converging to it and one can compute its image in any finite quotient group) as follows.

Take an endomorphism $\phi$ of $F_r$. The take a generator $x$ of $F_r$ and consider the element $\phi^{\omega}(x)$. This element is easily computed in the above senses and can be interesting. For example $\phi(x)=x^p$ is an endomorphism of the free group on one generator. The map $\phi^{\omega}$ takes $x$ to $x^{p^{\omega}}$.

Define $\psi\colon F_2\to F_2$ by $\psi(x)=[x,y]$ and $\psi(y)=y$ where $x,y$ are free generators. Then $\psi^{\omega}(x)$ is like an infinite iterated commutator $[x,y,y,...]$. Almeida denotes this element $[x,_{\omega}y]$ and observes a profinite group is pro-nilpotent iff it satisfies the profinite identity $[x,_{\omega} y]=1$.

There is also a 2-variable profinite identity defining pro-solvable groups, but this is more complicated and if memory serves relies on Thompson's classification of minimal non-sovable groups and maybe even the classification of finite simple groups.

Here are some Almeida surveys.

  1. http://cmup.fc.up.pt/cmup/jalmeida/preprints/finite-semigroups-and-dynamics-pdf.pdf
  2. http://cmup.fc.up.pt/cmup/jalmeida/preprints/Engel-CMUP.pdf
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Thank you for this illuminating answer (and comment)! –  Akhil Mathew Sep 29 '13 at 12:53
    
You are welcome. –  Benjamin Steinberg Sep 29 '13 at 13:28

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