  The true prosoluble completion $P\Cal S (\Gamma)$ of a group $\Gamma$ is the
inverse limit of the projective system of soluble quotients of $\Gamma$. Our
purpose is to describe examples and to point out some natural open problems. We
discuss a question of Grothendieck for profinite completions and its analogue
for true prosoluble and true pronilpotent completions.
  1. Introduction.
  2. Completion with respect to a directed set of normal subgroups.
  3. Universal property.
  4. Examples of directed sets of normal subgroups.
  5. True prosoluble completions.
  6. Examples.
  7. On the true prosoluble and the true pronilpotent analogues of
Grothendieck's problem.
