We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to Gromov’s well-known constructions, and include for example a ‘density model’ for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.

In this study of the behaviour of the number of conjugacy classes in finite $p$-groups using pro-$p$ groups, the conjugacy growth function $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ is introduced. It is proved that there are no infinite pro-$p$ groups of linear conjugacy growth (that is, there is no $c$ such that $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.