  According to Lazard, every p-adic Lie group contains an open pro-p subgroup
which is saturable. This can be regarded as the starting point of p-adic Lie
theory, as one can naturally associate to every saturable pro-p group G a Lie
lattice L(G) over the p-adic integers.
  Essential features of saturable pro-p groups include that they are
torsion-free and p-adic analytic. In the present paper we prove a converse
result in small dimensions: every torsion-free p-adic analytic pro-p group of
dimension less than p is saturable.
  This leads to useful consequences and interesting questions. For instance, we
give an effective classification of 3-dimensional soluble torsion-free p-adic
analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with
the use of Lazard's correspondence in the classification of finite p-groups of
small order.
