EDIT: Here is a more specific question.

Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ of $d$ copies of $G$; we equip $G^d$ with Haar measure $\mu$. Suppose that $\mu(S) > 0$. Does it follow that $S$ has non-empty interior?

(N.B. there are closed subsets of compact groups with positive measure and empty interior, e.g. you can put a fat Cantor set in the circle group; I just don't know if they can be solution sets of group equations.)


Let $G$ be a compact group (I am mainly interested in the profinite case). Pick a sequence of $d$ elements (where $d$ is either finite or $\omega$) independently at random (w.r.t. Haar measure) and look at the subgroup $\Gamma_d$ they generate. What can we say about $\Gamma_d$ almost surely, or at least with positive probability?

One context I've seen this question is with random finite generation of profinite groups: for a large class of topologically finitely generated profinite groups (including for instance all such groups that are pro-soluble), there exists $d$ such that $\Gamma_d$ is dense with positive probability. So we could condition on this and ask about properties of random dense $d$-generator subgroups. More generally, if $G$ is second-countable then $\Gamma_{\omega}$ is almost surely dense (since the generating set is almost surely dense). This doesn't say much about what $\Gamma_d$ looks like as a group, though.

It seems to me a natural tool to study $\Gamma_d$ is words/equations. Specifically, if $E$ is the set of word maps whose kernel in $G^{\omega}$ has measure $0$, then the generators will almost surely fail to satisfy any equation in $E$. If $E$ is all word maps that are not trivial on free groups, then $\Gamma_{\omega}$ is almost surely free. I suspect this covers quite a large class of profinite groups, giving a probabilistic argument for the existence of dense free subgroups (which may be difficult to construct explicitly from a profinite presentation).

What happens with equations where the solution set has positive measure? Do we have a 0-1 law for the probability that a variety contains a finite-index subgroup of $\Gamma_d$?

I am mainly looking for examples of previous work here, as I am sure someone else has had this idea before.


On random generations there are many results and you are probably familiar with them. Regrading your question, you should look at Belazs Szegedy's paper on the Nottingham group, http://blms.oxfordjournals.org/content/37/1/75. There are also results of Tsachik Gelander, Jonathan Barlev and Emanuel Breuillard on such questions.


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