  We give new upper bounds for the diameters of finite groups which do not
depend on a choice of generating set. Our method exploits the commutator
structure of certain profinite groups, in a fashion analogous to the
Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic
upper bounds for the diameters of finite quotients of: groups with an analytic
structure over a pro-$p$ domain (with exponent depending on the dimension);
Chevalley groups over a pro-$p$ domain (with exponent independent of the
dimension) and the Nottingham group of a finite field. We also discuss some
consequences of our results for random walks on groups.
