Let \(G\) be the group of infinite dimensional upper unitriangular matrices over a finite field. Then \(G\) is a profinite group and under a metric induced by the profinite topology is a complete metric space. In this paper the author considers ‘how many’ \(k\)-generator subgroups of \(G\) are free. Many authors have considered this question for other infinite groups.
The author proves that almost all \(k\)-generator subgroups are free by proving that the set of \(k\)-tuples \(\{x_1,\dots,x_k\}\) in \(G^k\) which do not generate a free group of rank \(k\) is a meagre set. The author does this by defining a countable subgroup \(H\) which is ‘full’ of \(k\)-generator subgroups, i.e. the intersection of \(H^k\) with any open ball in \(G^k\) contains a free subgroup of rank \(k\).
The author notes that \(G\) has many interesting non-free subgroups, for example the Nottingham group.